Improved estimates for the argument and zero-counting of Riemann zeta-function
(With an appendix by Andrew Fiori)

Chiara Bellotti School of Science
The University of New South Wales, Canberra, Australia c.bellotti@unsw.edu.au and Peng-Jie Wong National Sun Yat-Sen University
Department of Applied Mathematics
Kaohsiung City, Taiwan pjwong@math.nsysu.edu.tw

Abstract.

In this article, we improve the recent work of Hasanalizade, Shen, and Wong by establishing

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq 0.10076\log T+0.24460\log\log T+8.08292,

for every $T\geq e$ , where $N(T)$ is the number of non-trivial zeros $\rho=\beta+i\gamma$ , with $0<\gamma\leq T$ , of the Riemann zeta-function $\zeta(s)$ . The main source of improvement comes from implementing new subconvexity bounds for $\zeta(\sigma+it)$ on some $\sigma_{k}$ -lines inside the critical strip.

Key words and phrases:

Riemann zeta-function, zero-density estimates, explicit estimates

2020 Mathematics Subject Classification:

Primary 11M06, 11M26. Secondary 11Y35

P.J.W. is currently supported by the NSTC grant 111-2115-M-110-005-MY3. C.B. is partially supported by the AustMS WIMSIG Cheryl E. Praeger Travel Award. All the computation has been made using the Python program available on GitHub at [Bel].

1. Introduction

The main objective of this article is to improve the known estimates for the number of non-trivial zeros $\rho=\beta+i\gamma$ , with $0<\gamma\leq T$ , of the Riemann zeta-function $\zeta(s)$ . More precisely, we shall study

N(T)=\#\{\rho\in\mathbb{C}\mid\zeta(\rho)=0,\ 0<\beta<1,\ 0<\gamma\leq T\}

for $T>0$ . The first explicit bound for $N(T)$ was obtained by von Mangoldt [VM05], and it has been studied and improved since then. It shall be worthwhile mentioning that the importance of improving explicit bounds for $N(T)$ allows one to estimate sums over zeros of $\zeta(s)$ , it leads to the explicit bounds for the prime-counting functions $\pi(x)$ and $\psi(x)$ (see, e.g., [FK15]).

Our main theorem is the following estimate that improves the recent work of Hasanalizade, Shen, and Wong [HSW22]:

Theorem 1.1.

For every $T\geq e$ , we have

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq 0.10076\log T+0.24460\log\log T+8.08292

and

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq 0.11200\log T+0.12567\log\log T+3.77389.

For the convenience of the reader, writing

(1.1)

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq C_{1}\log T+C_{2}\log\log T+C_{3},

we summarise the advances that have been made in Table 1 below.¹¹1Regrettably, as pointed out in [HSW22], there is an error in [Tru14]. Also, there is a typo in [HSW22] regarding the constant $C_{3}$ . The value $9.3675$ is now substituted with $9.4925$ .

	$C_{1}$	$C_{2}$	$C_{3}$	$T_{0}$
von Mangoldt [VM05] (1905)	0.4320	1.9167	13.0788	28.5580
Grossmann [Gro13] (1913)	0.2907	1.7862	7.0120	50
Backlund [Bac16] (1918)	0.1370	0.4430	5.2250	200
Rosser [Ros41] (1941)	0.1370	0.4430	2.4630	2
Trudgian [Tru12] (2012)	0.1700	0	2.8730	$e$
Trudgian [Tru14] (2014)	0.1120	0.2780	3.3850	$e$
Platt–Trudgian [PT15] (2015)	0.1100	0.2900	3.165	$e$
Hasanalizade, Shen, and Wong [HSW22] (2022)	0.1038	0.2573	9.4925	$e$

Table 1. Previous explicit bounds for

N(T)

in (1.1)

Note that the second estimate in Theorem 1.1 is sharper than Trudgian’s bound [Tru14] for $T>378852$ .²²2This, together with Platt’s computation of $S(T)$ as stated in (1.2), assures that Trudgian’s bound [Tru14] and all the explicit results relying on it remain valid.

Similar to [HSW22], we derive Theorem 1.1 from the following general result.

Theorem 1.2.

Let $c,r,\eta$ be positive real numbers satisfying

-\frac{1}{2}<c-r<1-c<-\eta<\frac{1}{4}\leq\delta:=2c-\sigma_{1}-\frac{1}{2}<\frac{1}{2}<1+\eta<\sigma_{1}:=c+\frac{(c-1/2)^{2}}{r}<c+r

and $\theta_{1+\eta}\leq 2.1$ , where $\theta_{y}$ is defined in (3.32). Let $c_{1},c_{2},t_{0}$ defined in (2.2), $k_{1},k_{2},k_{3},t_{1}$ defined in (2.3), and denote with $n$ the largest $k$ that we consider when using (2.4).
Let $T_{0}\geq e$ be fixed. Then for any $T\geq T_{0}$ , we have

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq C_{1}\log T+\min\{C_{2}\log\log T+C_{3},C^{\prime}_{2}\log\log T+C^{\prime}_{3}\}

where the constants $C_{1}=C_{1}\left(c,r,\eta,k_{2},n\right)$ , $C_{2}=C_{2}\left(c,r,\eta,c_{2},k_{3},n\right)$ , $C^{\prime}_{2}=C^{\prime}_{2}\left(c,r,\eta,c_{2},k_{3},n\right)$ , $C_{3}=C_{3}(c,r,\eta,c_{1},c_{2},t_{0},k_{1},k_{2},k_{3},t_{1},n,T_{0})$ , $C^{\prime}_{3}=C^{\prime}_{3}(c,r,\eta,c_{1},c_{2},t_{0},k_{1},k_{2},k_{3},t_{1},n,T_{0})$ are defined in (3.44), (3.45), (3.46), (3.47), (3.48) respectively and some admissible values can be found in Table 2.

Remark.

The proof of Theorem 1.2 has its roots in the works of [Ben+21, Tru14, HSW22] and flourished by the following new inputs.
(i) We used various improved bounds for $\zeta(s)$ , including the estimates of Yang [Yan24] inside the critical strip. To implement these bounds, we further considered “ $n$ -splitting” of $[\frac{1}{2},1]$ in Section 3.1 ( $n=5$ gives nearly-optimal result for $C_{1}$ ). This is one of our key observations and leads us to refined estimates for $f_{N}$ and $F_{c,r}$ .

(ii) Instead of just using the trivial bound $|\zeta(\sigma+iT)|\geq\frac{\zeta(2\sigma)}{\zeta(\sigma)}$ for $\sigma>1$ , we provided an alternative method by applying a non-trivial estimate obtained by Leong recently (as stated in Lemma 2.2). In fact, the constants $C_{2}$ and $C_{3}$ in Theorem 1.2 are calculated via the trivial bound, while $C_{2}^{\prime}$ and $C_{3}^{\prime}$ are obtained by Leong’s bound. As may be noticed, Leong’s bound allows one to get a relatively small $C_{3}^{\prime}$ at the expense of a larger $C_{2}$ , which might be useful for certain applications.

(iii) We directly estimated $\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|$ (which is useful for most applications). This particularly reduces $C_{3}$ in the previous estimate of Hasanalizade, Shen, and Wong [HSW22] by roughly $\frac{1}{8}$ .

We shall remark that the constant $C_{1}=0.10076$ in Theorem 1.1 is the nearly-optimal value that one can obtain from Theorem 1.2 with the current knowledge on explicit estimates for $\zeta(s)$ . Also, it is possible to have smaller values of $C_{2},C^{\prime}_{2}$ or of $C_{3},C^{\prime}_{3}$ , at the expense of larger values for the other constants. Here, we provide two bounds with the smallest admissible $C_{2},C^{\prime}_{2}$ and $C_{3},C^{\prime}_{3}$ , respectively. (Particularly, the first estimate below is always sharper than Rosser’s bound [Ros41] for $T\geq e$ .)

Corollary 1.3.

The following estimates hold for $T\geq e$ :

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq 0.12355\log T+\min\{0.06782\log\log T+6.25781,0.62883\log\log T+2.05840\}

and

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq 0.16732\log T+\min\{0.17266\log\log T+1.96275,1.06148\log\log T+1.40212\}.

Another important consequence of Theorem 1.2 is the following general bound for the argument $\pi S(T)$ of the Riemann zeta-function along the $\frac{1}{2}$ -line. (Recall that $S(T)=\frac{1}{\pi}\Delta_{L}\arg\zeta(s)$ , where $L$ is the path formed by the straight line from $2$ to $2+iT$ and then to $\frac{1}{2}+iT$ ).

Theorem 1.4.

Under the same assumptions and notation of Theorem 1.2, the following estimate holds for every $T\geq T_{0}$ , with $T_{0}\geq e$ fixed:

|S(T)|\leq C_{1}\log T+\min\{C_{2}\log\log T+\tilde{C_{3}},C^{\prime}_{2}\log\log T+\tilde{C^{\prime}_{3}}\}

where the constants $C_{1}=C_{1}\left(c,r,\eta,k_{2},n\right)$ , $C_{2}=C_{2}\left(c,r,\eta,c_{2},k_{3},n\right)$ , $C^{\prime}_{2}=C^{\prime}_{2}\left(c,r,\eta,c_{2},k_{3},n\right)$ , $\tilde{C_{3}}=\tilde{C_{3}}(c,r,\eta,c_{1},c_{2},t_{0},k_{1},k_{2},k_{3},t_{1},n,T_{0})$ , $\tilde{C^{\prime}_{3}}=\tilde{C^{\prime}_{3}}(c,r,\eta,c_{1},c_{2},t_{0},k_{1},k_{2},k_{3},t_{1},n,T_{0})$ are defined in (3.44), (3.45), (3.46), (4.6), (4.7) respectively and some admissible values can be found in Table 2.

A sharper bound for $S(T)$ , up to certain given height, can be obtained using the database of non-trivial zeros of $\zeta(s)$ computed by Platt and made available at [Lmf]. More precisely, one has

(1.2)

|S(T)|\leq 2.5167

for $0\leq T\leq 30610046000.$ Hence, by Theorem 1.4 (with $T_{0}=30610046000$ and Table 2) and (1.2), we can get the following explicit estimate for $S(T)$ .

Corollary 1.5.

The following estimate holds for $T\geq e$ :

\left|S(T)\right|\leq 0.10076\log T+\min\{0.24460\log\log T+7.20792,1.13325\log\log T+1.50904\}.

Two final consequences of Theorems 1.2 and 1.4 are bounds for the number of zeros in a unit interval and a short interval. These types of estimates have several applications in number theory, such as providing an effective disproof of the Mertens conjecture [Pin87, RK23], improving the error term in the explicit version of the Riemann–von Mangoldt formula [Dud16], and consequently obtaining improvements related to primes between consecutive cubes and consecutive powers [JCH24]. More generally, these two types of estimates are useful for any problems that require an estimate for the sum over the zeros of $\zeta(s)$ restricted to a certain range.

Corollary 1.6.

Let $C_{1},C_{2},C^{\prime}_{2},\tilde{C}_{3},\tilde{C^{\prime}}_{3}$ be the constants defined in Theorem 1.2 and 1.4.
If $T+1>30610046000$ , then we have

		$\displaystyle N(T+1)-N(T)$
		$\displaystyle\leq\left(\frac{1}{2\pi}+2C_{1}\right)\log T+\min\{2C_{2}\log\log T+\mathscr{C}_{3},2C^{\prime}_{2}\log\log T+\mathscr{C^{\prime}}_{3}\}+\frac{1}{25T},$

where

	$\displaystyle\mathscr{C}_{3}$	$\displaystyle=2\tilde{C_{3}}+\frac{3}{4\pi}-\frac{1}{2\pi}\log(2\pi e),$
	$\displaystyle\mathscr{C^{\prime}}_{3}$	$\displaystyle=2\tilde{C^{\prime}_{3}}+\frac{3}{4\pi}-\frac{1}{2\pi}\log(2\pi e),$

and

	$\displaystyle\left(\frac{1}{\pi}-2.000001C_{1}\right)\log T-\min\{2C_{2}\log\log T+\mathscr{E},2C^{\prime}_{2}\log\log T+\mathscr{E^{\prime}}\}-\frac{1}{25(T-1)}$
	$\displaystyle\leq N(T+1)-N(T-1)$
	$\displaystyle\leq\left(\frac{1}{\pi}+2C_{1}\right)\log T+\min\{2C_{2}\log\log T+\mathscr{D}_{3},2C^{\prime}_{2}\log\log T+\mathscr{D^{\prime}}_{3}\}+\frac{1}{25(T-1)},$

where

	$\displaystyle\mathscr{D}_{3}$	$\displaystyle=2\tilde{C_{3}}+\frac{\log 3-\log(2\pi e)}{\pi},$
	$\displaystyle\mathscr{D^{\prime}}_{3}$	$\displaystyle=2\tilde{C^{\prime}_{3}}+\frac{\log 3-\log(2\pi e)}{\pi},$
	$\displaystyle\mathscr{E}$	$\displaystyle=2\tilde{C_{3}}+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e),$
	$\displaystyle\mathscr{E^{\prime}}$	$\displaystyle=2\tilde{C^{\prime}_{3}}+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e).$

Consequently, for $T+1>30610046000$ ,

		$\displaystyle N(T+1)-N(T)$
		$\displaystyle\leq\left(\frac{1}{2\pi}+0.20152\right)\log T+\min\{0.4892\log\log T+14.2030,2.2665\log\log T+2.8052\}+\frac{1}{25T}$

and

		$\displaystyle\left(\frac{1}{\pi}-0.201521\right)\log T-\min\{0.4892\log\log T+13.7851,2.2665\log\log T+2.3873\}-\frac{1}{25(T-1)}$
		$\displaystyle N(T+1)-N(T-1)$
		$\displaystyle\leq\left(\frac{1}{\pi}+0.20152\right)\log T+\min\{0.4892\log\log T+13.8623,2.2665\log\log T+2.4646\}+\frac{1}{25(T-1)}.$

In addition, for $3\leq T+1\leq 30610046000$ , we have

(1.3)

\frac{1}{2\pi}\log T-5.32592-\frac{1}{25T}\leq N(T+1)-N(T)\leq\frac{1}{2\pi}\log T+4.8405,

and

\frac{1}{\pi}\log T-5.66421-\frac{1}{25(T-1)}\leq N(T+1)-N(T-1)\leq\frac{1}{\pi}\log T+4.4798

Remark.

Note that the optimal constants that one could hope to get in front of the main terms in the upper bounds of $|N(T+1)-N(T)|$ and $|N(T+1)-N(T-1)|$ in Corollary 1.6 are $\frac{1}{2\pi}$ and $\frac{1}{\pi}$ respectively and it will be shown in Section 6. Furthermore, we omit the lower bound for $N(T+1)-N(T)$ when $T+1\geq 30610046000$ . Indeed, following the method used in Section 6 for $N(T+1)-N(T-1)$ results in a negative lower bound for $N(T+1)-N(T)$ , which is worse than the trivial non-negativity of $N(T+1)-N(T)$ .

2. Preliminaries

In this section, we recall some known results involving the Riemann zeta-function, which will be used throughout the proof of our theorems.
As already explained in [Tru14, HSW22], bounds of $\zeta(\sigma+it)$ when $\sigma=\frac{1}{2}$ and $\sigma=1$ play a crucial role. The sharpest known estimates for $\sigma=1$ and $t\geq 3$ are the following³³3Recently E. H. Qingyi and L.-P. Teo [QT24] found a new explicit bound for $|\zeta(1+it)|$ , which might lead to some small improvements.:

(2.1)

|\zeta(1+it)|\leq\left\{\begin{array}[]{lll}\min\{\log t,\frac{1}{2}\log t+1.93,\frac{1}{5}\log t+44.02\}&\text{if }3\leq t\leq e^{3070}\\ &\\ 1.731\frac{\log t}{\log\log t}&\text{if }e^{3070}<t\leq e^{3.69\cdot 10^{8}}\\ &\\ 58.096\log^{2/3}t&\text{if }t>e^{3.69\cdot 10^{8}}.\end{array}\right.

In order of appearance, the estimates are due to Patel [Pat22], Hiary–Leong–Yang [HLY24] and the last one is a direct consequence of Theorem 1 in [Bel24].
In order to apply Lemma 2.6, we actually need an estimate of the form

(2.2)

|\zeta(1+it)|\leq c_{1}(\log t)^{c_{2}}

for $t\geq t_{0}\geq e$ , where $c_{1},c_{2}$ are constants which depend on $t$ . Hence, for the middle range $\exp(3070)<t\leq\exp\left(3.69\cdot 10^{8}\right)$ we will use the slightly worse upper bound $|\zeta(1+it)|\leq 0.25\log t$ .
The sharpest known estimates for $|\zeta(\frac{1}{2}+it)|$ up to date are instead:

\left|\zeta\left(\frac{1}{2}+it\right)\right|\leq\left\{\begin{array}[]{ll}1.461&\text{ if }0\leq|t|\leq 3\\ \\ 0.618|t|^{1/6}\log|t|&\text{ if }3<|t|\leq\exp(105)\\ \\ 66.7t^{27/164}&\text{ if }|t|>\exp(105).\end{array}\right.

In order of appearance, these estimates are due to Hiary [Hia16], Hiary–Patel–Yang [HPY24], Patel–Yang [PY24]. We will denote

(2.3)

\left|\zeta\left(\frac{1}{2}+it\right)\right|\leq k_{1}t^{k_{2}}(\log t)^{k_{3}}

for $t\geq t_{1}\geq e$ , where $k_{1},k_{2},k_{3}$ are constants which depend on $t$ . Meanwhile, there exists another set of vertical lines inside the critical strip on which explicit bounds for $\zeta(s)$ are known. More precisely, Yang [Yan24] provided the following explicit estimate for $\left|\zeta\left(\sigma_{k}+it\right)\right|$ , with $\sigma_{k}:=1-k/\left(2^{k}-2\right)$ , for every $k\geq 4$ :

(2.4)

\left|\zeta\left(\sigma_{k}+it\right)\right|\leq 1.546t^{1/\left(2^{k}-2\right)}\log t,\quad t\geq 3.

For instance, substituting $k=4$ gives

|\zeta(5/7+it)|\leq 1.546t^{1/14}\log t.

The bound (2.4) will be the main innovative tool which will lead to improved values of both $C_{1}$ and $C_{2}$ in Theorem 1.1.

2.1. Other useful results

We recall other useful results which will be used to prove Theorem 1.1 and Theorem 1.2.
First of all, for $N\in\mathbb{N}$ , we consider the following function that will be fundamental in our argument:

f_{N}(s)=\frac{1}{2}\left(((s+iT-1)\zeta(s+iT))^{N}+((s-iT-1)\zeta(s-iT))^{N}\right).

Then, denoting with $D(c,r)$ the open disk centred at $c$ with radius $r$ , for any $N\in\mathbb{N}$ , we define

S_{N}(c,r)=\frac{1}{N}\sum_{z\in\mathscr{S}_{N}(D(c,r))}\log\frac{r}{|z-c|},

where $\mathscr{S}_{N}(D(c,r))$ is the set of zeros of $f_{N}(s)$ in $D(c,r)$ .

Lemma 2.1 ([HSW21] Prop. 3.5).

Let $c,r$ , and $\sigma_{1}$ be real numbers such that

c-r<\frac{1}{2}<1<c<\sigma_{1}<c+r.

Let $F_{c,r}:[-\pi,\pi]\rightarrow\mathbb{R}$ be an even function such that $F_{c,r}(\theta)\geq\frac{1}{N_{m}}\log\left|f_{N_{m}}\left(c+re^{i\theta}\right)\right|$ . Then there is an infinite sequence of natural numbers $\left(N_{m}\right)_{m=1}^{\infty}$ such that

\limsup_{m\rightarrow\infty}S_{N_{m}}(c,r)\leq\log\left(\frac{1}{\sqrt{(c-1)^{2}+T^{2}}}\frac{\zeta(c)}{\zeta(2c)}\right)+\frac{1}{\pi}\int_{0}^{\pi}F_{c,r}(\theta)d\theta.

Lemma 2.1 uses the following trivial lower bound for zeta when $\sigma>1$ :

(2.5)

|\zeta(\sigma+iT)|\geq\frac{\zeta(2\sigma)}{\zeta(\sigma)}.

However, recently Leong [Leo24] proved a non-trivial lower bound for $\zeta$ when $\sigma>1$ which does not depend on the real part $\sigma$ but only on the imaginary part $T$ .

Lemma 2.2 ([Leo24]).

The following estimate holds for $\sigma>1$ and $T>30610046000$ :

|\zeta(\sigma+iT)|>\frac{1}{2.0945\log T}.

If instead of the trivial bound (2.5) we use Lemma 2.2 inside the proof of Lemma 2.1, we get the following result.

Lemma 2.3.

Under the same assumption as in Lemma 2.1, if $T>30610046000$ there is an infinite sequence of natural numbers $\left(N_{m}\right)_{m=1}^{\infty}$ such that

\limsup_{m\rightarrow\infty}S_{N_{m}}(c,r)\leq\log\left(\frac{2.0945\log T}{\sqrt{(c-1)^{2}+T^{2}}}\right)+\frac{1}{\pi}\int_{0}^{\pi}F_{c,r}(\theta)d\theta.

Backlund’s trick is another important tool we will use for the proof of Theorem 1.1. To end this section, we recall the following version of Backlund’s trick (see [HSW21, Prop. 3.7]), a tool according to which if $f_{N}(\sigma)$ has zeros in $\left[\frac{1}{2},\sigma_{1}\right]$ , then there are zeros in $\left[1-\sigma_{1},\frac{1}{2}\right]$ .

Lemma 2.4 (Backlund’s trick).

Let $c$ and $r$ be real numbers. Set

\sigma_{1}=c+\frac{(c-1/2)^{2}}{r}\quad\text{ and }\quad\delta=2c-\sigma_{1}-\frac{1}{2}.

If $1<c<r$ and $0<\delta<\frac{1}{2}$ , then

\left|\arg((\sigma+iT-1)\zeta(\sigma+iT))|_{\sigma=\sigma_{1}}^{1/2}\right|\leq\frac{\pi S_{N}(c,r)}{2\log(r/(c-1/2))}+\frac{E(T,\delta)}{2}+\frac{\pi}{N}+\frac{\pi}{2N}+\frac{\pi}{4},

where $E(T,\delta)$ is defined in [Ben+21, p. 1463].

We also recall the estimate for $E(T,\delta)/\pi$ due to [Ben+21, Lemma 3.4].

Lemma 2.5 ([Ben+21]).

For $0\leq\delta_{1}\leq d<9/2$ and $T\geq 5/7$ , one has

0<E\left(T,\delta_{1}\right)\leq E(T,d)

Also, for $d\in\left[\frac{1}{4},\frac{5}{8}\right]$ and $T\geq 5/7$ , one has

\frac{E(T,d)}{\pi}\leq\frac{640d-112}{1536(3T-1)}+\frac{1}{2^{10}}.

To conclude the section, we mention the Phragmén–Lindelöf principle, as stated in [Tru14, Lemma 3], which will be used to construct a suitable function $F_{c,r}(\theta)$ , given Lemmata 2.1, 2.3, and 2.4.

Lemma 2.6 (Phragmén-Lindelöf principle).

Let $a,b,Q$ be real numbers such that $b>$ $a$ and $Q+a>1$ . Let $f(s)$ be a holomorphic function on the strip $a\leq\mathfrak{Re}(s)\leq b$ such that

|f(s)|<C\exp\left(e^{k|t|}\right)

for some $C>0$ and $0<k<\frac{\pi}{b-a}$ . Suppose, further, that there are $A,B,\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\geq 0$ such that $\alpha_{1}\geq\beta_{1}$ and

|f(s)|\leq\left\{\begin{array}[]{ll}A|Q+s|^{\alpha_{1}}(\log|Q+s|)^{\alpha_{2}}&\text{ for }\mathfrak{Re}(s)=a;\\ B|Q+s|^{\beta_{1}}(\log|Q+s|)^{\beta_{2}}&\text{ for }\mathfrak{Re}(s)=b.\end{array}\right.

Then for $a\leq\mathfrak{Re}(s)\leq b$ , one has

|f(s)|\leq\left\{A|Q+s|^{\alpha_{1}}(\log|Q+s|)^{\alpha_{2}}\right\}^{\frac{b-\mathfrak{Re}(s)}{b-a}}\left\{B|Q+s|^{\beta_{1}}(\log|Q+s|)^{\beta_{2}}\right\}^{\frac{\mathfrak{Re}(s)-a}{b-a}}.

3. Proof of Theorem 1.2

As in [Tru14, HSW22], we will work with the completed Riemann zeta-function $\xi(s)$ , which is defined by

\xi(s)=s(s-1)\gamma(s)\zeta(s)\quad\text{with}\quad\gamma(s)=\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right).

It is widely known that the function $\xi(s)$ is an entire function of order 1 , which satisfies the functional equation

\xi(s)=\xi(1-s).

Furthermore, as in [HSW22], instead of working directly with the quantity $N(T)$ , we will work with the quantity $N_{\mathbb{Q}}(T)$ which is defined as

N_{\mathbb{Q}}(T)=\#\{\rho\in\mathbb{C}|\zeta(\rho)=0,0<\beta<1,|\gamma\mid\leq T\}

for $T\geq 0$ and it is related to the quantity $N(T)$ by the equality $N_{\mathbb{Q}}(T)=2N(T)$ .
Following [HSW22], given a parameter $\sigma_{1}>1$ we will choose later, we consider the rectangle $\mathscr{R}$ with vertices $\sigma_{1}-iT$ , $\sigma_{1}+iT$ , $1-\sigma_{1}+iT$ , and $1-\sigma_{1}-iT$ , and, since $\xi(s)$ is entire, we apply the argument principle, getting

N_{\mathbb{Q}}(T)=\frac{1}{2\pi}\Delta_{\mathscr{R}}\arg\xi(s).

As per [HSW22, eq. (2.8)], the previous expression for $N_{\mathbb{Q}}(T)$ is equivalent to

(3.1)

N_{\mathbb{Q}}(T)=\frac{2}{\pi}\arctan(2T)+g(T)+\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{1}{4}+\frac{2}{\pi}\Delta_{\mathscr{C}_{0}}\arg((s-1)\zeta(s)),

where

g(T)=\frac{2}{\pi}\mathfrak{Im}\log\Gamma\left(\frac{1}{4}+i\frac{T}{2}\right)-\frac{T}{\pi}\log\left(\frac{T}{2e}\right)+\frac{1}{4},

and $\mathscr{C}_{0}$ is the part of the contour of $\mathscr{R}$ in the region $\mathfrak{Im}(s)\geq 0$ and $\mathfrak{Re}(s)\geq\frac{1}{2}$ . The estimate $|g(T)|\leq 1/(25T)$ is given by [Ben+21, Prop. 3.2] and holds for every $T\geq 5/7$ . Then, denoting with $\mathscr{C}_{1}$ the vertical line from $\sigma_{1}$ to $\sigma_{1}+iT$ and with $\mathscr{C}_{2}$ the horizontal line from $\sigma_{1}+iT$ to $\frac{1}{2}+iT$ , one has

	$\displaystyle\Delta_{\mathscr{C}_{0}}\arg((s-1)\zeta(s))$	$\displaystyle=\Delta_{\mathscr{C}_{1}}\arg((s-1)\zeta(s))+\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))$
		$\displaystyle=\arctan\left(\frac{T}{\sigma_{1}-1}\right)+\Delta_{\mathscr{C}_{1}}\arg\zeta(s)+\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))$

and for $\sigma_{1}>1$ ,

\left|\Delta_{\mathscr{C}_{1}}\arg\zeta(s)\right|=\left|\arg\zeta\left(\sigma_{1}+iT\right)\right|\leq\left|\log\zeta\left(\sigma_{1}+iT\right)\right|\leq\log\zeta\left(\sigma_{1}\right).

Furthermore, we know that [HSW22, eq. (5.5)]

(3.2)

S(T)=\frac{1}{\pi}\Delta_{\mathscr{C}_{0}}\arg\zeta(s)=\frac{1}{2}\left(N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)+\frac{1}{4}-g(T)-2\right)

and

(3.3)

S(T)=\frac{1}{\pi}\Delta_{\mathscr{C}_{0}}\arg\zeta(s)=\frac{1}{\pi}\Delta_{\mathscr{C}_{1}}\arg\zeta(s)+\frac{1}{\pi}\Delta_{\mathscr{C}_{2}}\arg(s-1)\zeta(s)-\frac{1}{\pi}\Delta_{\mathscr{C}_{2}}\arg(s-1).

Let $T\geq T_{0}$ fixed. Instead of estimating the usual quantity

\left|N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)+\frac{1}{4}\right|

as in [HSW22] (or $\frac{1}{8}$ instead of $\frac{1}{4}$ if we work with $N(T)$ instead of $N_{\mathbb{Q}}(T)$ as in [Tru14]), we aim to estimate

\left|N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)\right|,

which is the quantity that is usually required in the majority of applications.

Remark.

This choice will allow us to save an extra $0.25$ in the final constant $C_{3}$ in Theorem 1.1.

One has

(3.4)			$\displaystyle\left\|N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$
			$\displaystyle\leq\left\|\frac{1}{4}-\frac{2}{\pi}\arctan\left(\frac{T}{\sigma_{1}-1}\right)-\frac{2}{\pi}\arctan(2T)\right\|+\|g(T)\|+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
			$\displaystyle\leq\left\|\frac{1}{4}-\frac{2}{\pi}\arctan\left(\frac{T}{\sigma_{1}-1}\right)-\frac{2}{\pi}\arctan(2T)\right\|+\frac{1}{25T}+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
			$\displaystyle=\|h(c,r,T)\|+\frac{1}{25T}+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|,$

where

h(c,r,T)=\frac{1}{4}-\frac{2}{\pi}\arctan\left(\frac{T}{\sigma_{1}-1}\right)-\frac{2}{\pi}\arctan(2T).

Since $\sigma_{1}>1$ (it will be approximately $\sigma_{1}\sim 1.25$ ), $T$ quite large ( $T>30610046000$ ), and being the function $\arctan$ increasing for $T>0$ , we can estimate $|h(c,r,T)|$ as

(3.5)

|h(c,r,T)|\leq\left|\frac{2}{\pi}\arctan(4T)+\frac{2}{\pi}\arctan(2T)-\frac{1}{4}\right|\leq 2-\frac{1}{4}=\frac{7}{4}.

Now, we will focus on improving the bound for

\left|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right|,

for which the main new ideas which lead to improvements are used.

3.1. Convexity/subconvexity bounds and generalisation to $n$ -splitting of $[\frac{1}{2},1]$

While outside the range $[\frac{1}{2},1]$ we consider the same cases as in [HSW22], we will split the interval $[\frac{1}{2},1]$ in $n$ sub-intervals using the lines corresponding to (2.4). As already mentioned, this splitting will be the main tool which allows us to get an improved value of $C_{1}$ .
More precisely, the first sub-interval is of the form $[\frac{1}{2},\sigma_{4}]$ , the last interval will be of the form $[\sigma_{n+4},1]$ and the intervals in the middle will be of the form $[\sigma_{4+h},\sigma_{4+h+1}]$ , where $0\leq h\leq n-1$ and, for every $k$ , $\sigma_{k}$ is defined as

\sigma_{k}:=1-k/(2^{k}-2).

A careful analysis shows that the nearly-optimal value for $n$ is $5$ . Indeed, a higher number of subintervals of the form $[\sigma_{4+h},\sigma_{4+h+1}]$ would lead to an improvement on $C_{1}$ only at the seventh decimal place, while it would cause a worse constant $C_{3}$ , in which a factor of containing $\log(1.546)$ (multiplied by other quantities) appears in $C_{3}$ for each interval $[\sigma_{4+h},\sigma_{4+h+1}]$ we are considering.
Following [HSW22], we start estimating $|\zeta(\sigma+it)|$ in each of the intervals for $\sigma$ we are considering.

•

Case $\sigma\geq 1+\eta$ . The trivial bound for the zeta-function immediately implies $|\zeta(s)|\leq\zeta(\sigma).$

•

Case $1\leq\sigma\leq 1+\eta$ . From (2.1) and (2.2) it follows that there is $Q_{0}>0$ depending on $c_{1},c_{2},t_{0}$ such that

(3.6)

|(1+it-1)\zeta(1+it)|\leq c_{1}\left|Q_{0}+(1+it)\right|\left(\log\left|Q_{0}+(1+it)\right|\right)^{c_{2}}

for all $t$ . Thus, Lemma 2.6 implies that for $1\leq\sigma\leq 1+\eta$ ,

|(s-1)\zeta(s)|\leq\left(c_{1}\left|Q_{0}+s\right|\left(\log\left|Q_{0}+s\right|\right)^{c_{2}}\right)^{\frac{1+\eta-\sigma}{\eta}}\left(\zeta(1+\eta)\left|Q_{0}+s\right|\right)^{\frac{\sigma-1}{\eta}},

and hence

|\zeta(s)|\leq\frac{1}{|s-1|}\left(c_{1}\left|Q_{0}+s\right|\left(\log\left|Q_{0}+s\right|\right)^{c_{2}}\right)^{\frac{1+\eta-\sigma}{\eta}}\left(\zeta(1+\eta)\left|Q_{0}+s\right|\right)^{\frac{\sigma-1}{\eta}}.

•

Case $\sigma_{n+4}\leq\sigma\leq 1$ . By (2.4), there exists $Q_{n+4}>0$ depending on $n$ such that

(3.7)			$\displaystyle\|(\sigma_{n+4}+it-1)\zeta(\sigma_{n+4}+it)\|$
(3.7)			$\displaystyle\leq 1.546\|Q_{n+4}+(\sigma_{n+4}+it)\|^{\frac{1}{2^{n+4}-2}+1}(\log\|Q_{n+4}+(\sigma_{n+4}+it)\|)$

Lemma 2.6 and (3.6) imply that for $\sigma_{n+4}\leq\sigma\leq 1$ ,

	$\displaystyle\|\zeta(s)\|$	$\displaystyle\leq\frac{1}{\|s-1\|}\left(1.546\|Q_{0,n+4}+s\|^{\frac{1}{2^{n+4}-2}+1}(\log\|Q_{0,n+4}+s\|)\right)^{\frac{1-\sigma}{1-\sigma_{n+4}}}$
		$\displaystyle\times\left(c_{1}\left\|Q_{0,n+4}+s\right\|\left(\log\left\|Q_{0,n+4}+s\right\|\right)^{c_{2}}\right)^{\frac{\sigma-\sigma_{n+4}}{1-\sigma_{n+4}}},$

where $Q_{0,n+4}=\max\{Q_{0},Q_{n+4}\}$ .

•

Case $\sigma_{4+h}\leq\sigma\leq\sigma_{4+h+1}$ with $0\leq h\leq n-1$ . By (2.4), for every fixed $h$ , there exist $Q_{4+h},Q_{5+h}>0$ depending on $h$ so that

(3.8)			$\displaystyle\|(\sigma_{h+4}+it-1)\zeta(\sigma_{h+4}+it)\|$
(3.8)			$\displaystyle\leq 1.546\|Q_{h+4}+(\sigma_{h+4}+it)\|^{\frac{1}{2^{h+4}-2}+1}(\log\|Q_{h+4}+(\sigma_{h+4}+it)\|)$

and

(3.9)			$\displaystyle\|(\sigma_{h+5}+it-1)\zeta(\sigma_{h+5}+it)\|$
(3.9)			$\displaystyle\leq 1.546\|Q_{h+5}+(\sigma_{h+5}+it)\|^{\frac{1}{2^{h+5}-2}+1}(\log\|Q_{h+5}+(\sigma_{h+5}+it)\|).$

Hence, by Lemma 2.6, we have

	$\displaystyle\|\zeta(s)\|\leq\frac{1}{\|s-1\|}\left(1.546\|Q_{4+h,5+h}+s\|^{\frac{1}{2^{h+4}-2}+1}(\log\|Q_{4+h,5+h}+s\|)\right)^{\frac{\sigma_{5+h}-\sigma}{\sigma_{5+h}-\sigma_{h+4}}}$
	$\displaystyle\qquad\times\left(1.546\|Q_{4+h,5+h}+s\|^{\frac{1}{2^{h+5}-2}+1}(\log\|Q_{4+h,5+h}+s\|)\right)^{\frac{\sigma-\sigma_{h+4}}{\sigma_{h+5}-\sigma_{h+4}}},$

where $Q_{4+h,5+h}=\max\{Q_{4+h},Q_{5+h}\}$ .

Remark.

When $h=n-1$ , $Q_{h+5}=Q_{n+4}$ , with $Q_{n+4}$ given in (3.7).

•

Case $1/2\leq\sigma\leq\sigma_{4}$ . By (2.3), there is a $Q_{1}>0$ depending on $k_{1},k_{2},k_{3},t_{1}$ such that

(3.10)

\displaystyle\left|\left(\frac{1}{2}+it-1\right)\zeta\left(\frac{1}{2}+it\right)\right|\leq k_{1}\left|Q_{1}+\left(\frac{1}{2}+it\right)\right|^{k_{2}+1}\left(\log\left|Q_{1}+\left(\frac{1}{2}+it\right)\right|\right)^{k_{3}}.

Hence, Lemma 2.6 and (3.8) with $h=0$ imply that

	$\displaystyle\|\zeta(s)\|$
	$\displaystyle\leq\frac{1}{\|s-1\|}\left(k_{1}\left\|Q_{2}+s\right\|^{k_{2}+1}\left(\log\left\|Q_{2}+s\right\|\right)^{k_{3}}\right)^{\frac{\sigma_{4}-\sigma}{\sigma_{4}-\frac{1}{2}}}\left(1.546\|Q_{2}+s\|^{\frac{1}{2^{4}-2}+1}(\log\|Q_{2}+s\|)\right)^{\frac{\sigma-\frac{1}{2}}{\sigma_{4}-\frac{1}{2}}},$

where $Q_{2}=\max\{Q_{1},Q_{4+0}\}$ .

•

Case $0\leq\sigma\leq\frac{1}{2}$ . Following [HSW22], there exists $Q_{3}>0$ such that

(3.11)

\left|\zeta\left(\frac{1}{2}+it\right)\right|\leq k_{1}\left|Q_{3}+\left(\frac{1}{2}+it\right)\right|^{k_{2}}\left(\log\left|Q_{3}+\left(\frac{1}{2}+it\right)\right|\right)^{k_{3}}

for all $t$ and $Q_{10}\geq 1$ such that

(3.12)

|\zeta(0+it)|\leq\frac{c_{1}}{\sqrt{2\pi}}\left|Q_{10}+it\right|^{\frac{1}{2}}\left(\log\left|Q_{10}+it\right|\right)^{c_{2}}.

Hence, by Lemma 2.6 one has

(3.13)

|\zeta(s)|\leq\left(\frac{c_{1}}{\sqrt{2\pi}}\left|Q_{11}+s\right|^{\frac{1}{2}}\left(\log\left|Q_{11}+s\right|\right)^{c_{2}}\right)^{1-2\sigma}\left(k_{1}\left|Q_{11}+s\right|^{k_{2}}\left(\log\left|Q_{11}+s\right|\right)^{k_{3}}\right)^{2\sigma}

where $Q_{11}=\max\{Q_{3},Q_{10}\}$ .

•

Case $-\eta\leq\sigma\leq 0$ . As in [HSW22], we have

(3.14)

|\zeta(s)|\leq\left(\frac{1}{(2\pi)^{\frac{1}{2}+\eta}}\zeta(1+\eta)\left|Q_{10}+s\right|^{\frac{1}{2}+\eta}\right)^{\frac{-\sigma}{\eta}}\left(\frac{c_{1}}{\sqrt{2\pi}}\left|Q_{10}+s\right|^{\frac{1}{2}}\left(\log\left|Q_{10}+s\right|\right)^{c_{2}}\right)^{\frac{\sigma+\eta}{\eta}}.

•

Case $\sigma\leq-\eta$ . We shall use the same estimate as in [HSW22]:

(3.15)

|\zeta(s)|\leq\zeta(1-\sigma)\left(\frac{1}{2\pi}\right)^{\frac{1}{2}-\sigma}(|1+s-[\sigma]|)^{\frac{1}{2}+[\sigma]-\sigma}\left(\prod_{j=1}^{-[\sigma]}|s+j-1|\right).

3.2. Estimating $\frac{1}{N}\log|f_{N}(s)|$

Given the function

(3.16)

f_{N}(s)=\frac{1}{2}\left(((s+iT-1)\zeta(s+iT))^{N}+((s-iT-1)\zeta(s-iT))^{N}\right),

we want to bound

(3.17)

\frac{1}{N}\log\left|f_{N}(s)\right|

inside the different ranges we considered in the previous subsection.

•

Case $\sigma\geq 1+\eta$ . As per [HSW22], we have the bound

(3.18) $\frac{1}{N}\log\left|f_{N}(s)\right|\leq\frac{1}{2}\log\left((\sigma-1)^{2}+(|t|+T)^{2}\right)+\log\zeta(\sigma).$

•

Case $1\leq\sigma\leq 1+\eta$ . Following [HSW22], we use

(3.19)		$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq\frac{1+\eta-\sigma}{\eta}\log\frac{c_{1}}{2^{c_{2}}}+\frac{\sigma-1}{\eta}\log\zeta(1+\eta)+\frac{1}{2}\log\left(\left(Q_{0}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
(3.19)			$\displaystyle+\frac{c_{2}(1+\eta-\sigma)}{\eta}\log\log\left(\left(Q_{0}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

•

Case $\sigma_{n+4}\leq\sigma\leq 1$ . Observe that

	$\displaystyle\|f_{N}(s)\|$	$\displaystyle\leq\left(1.546((Q_{0,n+4}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{2^{n+4}-1}{2(2^{n+4}-2)}}\log(\sqrt{(Q_{0,n+4}+\sigma)^{2}+(\|t\|+T)^{2}})\right)^{\frac{N(1-\sigma)}{1-\sigma_{n+4}}}$
		$\displaystyle\times\left(c_{1}((Q_{0,n+4}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{1}{2}}\left(\log(\sqrt{(Q_{0,n+4}+\sigma)^{2}+(\|t\|+T)^{2}})\right)^{c_{2}}\right)^{\frac{N(\sigma-\sigma_{n+4})}{1-\sigma_{n+4}}}.$

Hence, taking the logarithms of both sides and dividing by $N$ , we obtain

	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|\leq\frac{(1-\sigma)}{1-\sigma_{n+4}}(\log 1.546)+\frac{(\sigma-\sigma_{n+4})}{1-\sigma_{n+4}}\log c_{1}-\left(\frac{(1-\sigma)}{(1-\sigma_{n+4})}+\frac{c_{2}(\sigma-\sigma_{n+4})}{(1-\sigma_{n+4})}\right)\log 2$
	$\displaystyle+\left(\frac{(2^{n+4}-1)(1-\sigma)}{2(2^{n+4}-2)(1-\sigma_{n+4})}+\frac{(\sigma-\sigma_{n+4})}{2(1-\sigma_{n+4})}\right)\log\left(\left(Q_{0,n+4}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle+\left(\frac{(1-\sigma)}{(1-\sigma_{n+4})}+\frac{c_{2}(\sigma-\sigma_{n+4})}{(1-\sigma_{n+4})}\right)\log\log\left(\left(Q_{0,n+4}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

•

Case $\sigma_{4+h}\leq\sigma\leq\sigma_{4+h+1}$ , where $0\leq h\leq n-1$ . It follows from

	$\displaystyle\|f_{N}(s)\|$
	$\displaystyle\leq\left(1.546((Q_{4+h,5+h}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{2^{h+4}-1}{2(2^{h+4}-2)}}\log(\sqrt{(Q_{4+h,5+h}+\sigma)^{2}+(\|t\|+T)^{2}})\right)^{\frac{N(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}}$
	$\displaystyle\times(1.546((Q_{4+h,5+h}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{2^{h+5}-1}{2(2^{h+5}-2)}}(\log(\sqrt{(Q_{4+h,5+h}+\sigma)^{2}+(\|t\|+T)^{2}})))^{\frac{N(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}}$

that

	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$
	$\displaystyle\leq\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)(\log 1.546)-\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)\log 2$
	$\displaystyle+\left(\frac{(2^{h+4}-1)(\sigma_{5+h}-\sigma)}{2(2^{h+4}-2)(\sigma_{5+h}-\sigma_{h+4})}+\frac{(2^{h+5}-1)(\sigma-\sigma_{h+4})}{2(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}\right)\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle+\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)\log\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle=\log 1.546-\log 2+\log\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle+\left(\frac{(2^{h+4}-1)(\sigma_{5+h}-\sigma)}{2(2^{h+4}-2)(\sigma_{5+h}-\sigma_{h+4})}+\frac{(2^{h+5}-1)(\sigma-\sigma_{h+4})}{2(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}\right)\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

•

Case $1/2\leq\sigma\leq\sigma_{4}$ . One has

	$\displaystyle\|f_{N}(s)\|$	$\displaystyle\leq\left(k_{1}((Q_{2}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{k_{2}+1}{2}}\log(\sqrt{(Q_{2}+\sigma)^{2}+(\|t\|+T)^{2}})^{k_{3}}\right)^{\frac{N(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}}$
		$\displaystyle\times(1.546((Q_{2}+\sigma)^{2}+(\|t\|+T)^{2})^{\frac{15}{28}}(\log(\sqrt{(Q_{2}+\sigma)^{2}+(\|t\|+T)^{2}})))^{\frac{N(\sigma-\frac{1}{2})}{\sigma_{4}-\frac{1}{2}}}$

and thus

	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq\frac{(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}(\log k_{1})+\frac{(\sigma-\frac{1}{2})}{\sigma_{4}-\frac{1}{2}}(\log 1.546)-\left(\frac{k_{3}(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}+\frac{(\sigma-\frac{1}{2})}{(\sigma_{4}-\frac{1}{2})}\right)\log 2$
		$\displaystyle+\left(\frac{(k_{2}+1)(\sigma_{4}-\sigma)}{2(\sigma_{4}-\frac{1}{2})}+\frac{15(\sigma-\frac{1}{2})}{28(\sigma_{4}-\frac{1}{2})}\right)\log\left(\left(Q_{2}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\left(\frac{k_{3}(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}+\frac{(\sigma-\frac{1}{2})}{(\sigma_{4}-\frac{1}{2})}\right)\log\log\left(\left(Q_{2}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

•

Case $0\leq\sigma\leq 1/2$ . As in [HSW22], we have

(3.20)	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq(1-2\sigma)\log\left(\frac{c_{1}}{2^{c_{2}+\frac{1}{2}}\sqrt{\pi}}\right)+2\sigma\log\frac{k_{1}}{2^{k_{3}}}+\frac{1}{2}\log\left((\sigma-1)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\frac{1-2\sigma+4k_{2}\sigma}{4}\log\left(\left(Q_{11}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\left(c_{2}(1-2\sigma)+2k_{3}\sigma\right)\log\log\left(\left(Q_{11}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$

•

Case $-\eta\leq\sigma\leq 0$ . By [HSW22], we know

(3.21)	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq-\frac{\sigma}{\eta}\log\frac{1}{(2\pi)^{\frac{1}{2}+\eta}}-\frac{\sigma}{\eta}\log(1+\eta)+\frac{\sigma+\eta}{\eta}\log\frac{c_{1}}{\sqrt{2\pi}}-\frac{\sigma+\eta}{\eta}c_{2}\log 2$
		$\displaystyle+\frac{1}{2}\log\left((\sigma-1)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\left(-\frac{\sigma(1+2\eta)}{4\eta}+\frac{\sigma+\eta}{4\eta}\right)\log\left(\left(Q_{10}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\frac{\sigma+\eta}{\eta}c_{2}\log\log\left(\left(Q_{10}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

•

Case $\sigma\leq-\eta$ . Finally, as per [HSW22], we use the bound

(3.22)	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq\log\zeta(1-\sigma)+\frac{1}{2}\log\left((\sigma-1)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\frac{2\sigma-1}{2}\log 2\pi+\frac{(1-2\sigma+2[\sigma])}{4}\log\left((1+\sigma-[\sigma])^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\frac{1}{2}\sum_{j=1}^{-[\sigma]}\log\left((\sigma+j-1)^{2}+(\|t\|+T)^{2}\right).$

3.3. Defining $F_{c,r}(\theta)$

We start recalling some auxiliary functions already defined in [HSW22] which will appear in the definition of $F_{c,r}(\theta)$ . For $\theta\in$ $[-\pi,\pi]$ , we let $\sigma=c+r\cos\theta$ , with $c-r>-\frac{1}{2}$ , and $t=r\sin\theta$ . We define

(3.23)

L_{j}(\theta)=\log\frac{(j+c+r\cos\theta)^{2}+(|r\sin\theta|+T)^{2}}{T^{2}}

and

(3.24)

M_{j}(\theta)=\log\log\left((j+c+r\cos\theta)^{2}+(|r\sin\theta|+T)^{2}\right)-\log\log\left(T^{2}\right).

•

If $\sigma\geq 1+\eta$ , as per [HSW22] we define

$F_{c,r}(\theta)=\frac{1}{2}L_{-1}(\theta)+\log T+\log\zeta(\sigma).$

•

For $1\leq\sigma\leq 1+\eta$ , as per [HSW22]

(3.25)		$\displaystyle F_{c,r}(\theta)$	$\displaystyle=\frac{1+\eta-\sigma}{\eta}\log c_{1}+\frac{\sigma-1}{\eta}\log\zeta(1+\eta)+\frac{1}{2}L_{Q_{0}}(\theta)+\log T$
(3.25)			$\displaystyle+\frac{c_{2}(1+\eta-\sigma)}{\eta}M_{Q_{0}}(\theta)+\frac{c_{2}(1+\eta-\sigma)}{\eta}\log\log T.$

•

If $\sigma_{n+4}\leq\sigma\leq 1$ , then we define

	$\displaystyle F_{c,r}(\theta)$	$\displaystyle=\frac{(1-\sigma)}{1-\sigma_{n+4}}(\log 1.546)+\frac{(\sigma-\sigma_{n+4})}{1-\sigma_{n+4}}\log c_{1}$
		$\displaystyle+\left(\frac{(2^{n+4}-1)(1-\sigma)}{(2^{n+4}-2)(1-\sigma_{n+4})}+\frac{(\sigma-\sigma_{n+4})}{(1-\sigma_{n+4})}\right)\left(\frac{L_{Q_{0,n+4}}(\theta)}{2}+\log T\right)$
		$\displaystyle+\left(\frac{(1-\sigma)}{(1-\sigma_{n+4})}+\frac{c_{2}(\sigma-\sigma_{n+4})}{(1-\sigma_{n+4})}\right)(M_{Q_{0,n+4}}(\theta)+\log\log T).$

•

When $\sigma_{4+h}\leq\sigma\leq\sigma_{4+h+1}$ , where $0\leq h\leq n-1$ , then

	$\displaystyle F_{c,r}(\theta)$	$\displaystyle=\log 1.546+(M_{Q_{4+h,5+h}}(\theta)+\log\log T)$
		$\displaystyle+\left(\frac{(2^{h+4}-1)(\sigma_{5+h}-\sigma)}{(2^{h+4}-2)(\sigma_{5+h}-\sigma_{h+4})}+\frac{(2^{h+5}-1)(\sigma-\sigma_{h+4})}{(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}\right)\left(\frac{L_{Q_{4+h,5+h}}(\theta)}{2}+\log T\right).$

•

If $1/2\leq\sigma\leq\sigma_{4}$ , we define

	$\displaystyle F_{c,r}(\theta)$	$\displaystyle=\frac{(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}(\log k_{1})+\frac{(\sigma-\frac{1}{2})}{\sigma_{4}-\frac{1}{2}}(\log 1.546)$
		$\displaystyle+\left(\frac{(k_{2}+1)(\sigma_{4}-\sigma)}{(\sigma_{4}-\frac{1}{2})}+\frac{15(\sigma-\frac{1}{2})}{14(\sigma_{4}-\frac{1}{2})}\right)\left(\frac{L_{Q_{2}}(\theta)}{2}+\log T\right)$
		$\displaystyle+\left(\frac{k_{3}(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}+\frac{(\sigma-\frac{1}{2})}{(\sigma_{4}-\frac{1}{2})}\right)(M_{Q_{2}}(\theta)+\log\log T).$

•

For $0\leq\sigma\leq 1/2$ , as in [HSW22],

(3.26)		$\displaystyle F_{c,r}(\theta)$	$\displaystyle=(1-2\sigma)\log\frac{c_{1}}{\sqrt{2\pi}}+2\sigma\log k_{1}+\frac{1}{2}L_{-1}(\theta)+\log T$
(3.26)			$\displaystyle+\frac{1-2\sigma+4k_{2}\sigma}{2}\left(\frac{L_{Q_{11}}(\theta)}{2}+\log T\right)+\left(c_{2}(1-2\sigma)+2k_{3}\sigma\right)\left(M_{Q_{11}}(\theta)+\log\log T\right).$

•

If $-\eta\leq\sigma\leq 0$ then as in [HSW22]

(3.27)		$\displaystyle F_{c,r}(\theta)$	$\displaystyle=-\frac{\sigma}{\eta}\log\frac{1+\eta}{c_{1}(2\pi)^{\eta}}+\log\frac{c_{1}}{\sqrt{2\pi}}+\frac{1}{2}L_{-1}(\theta)+\log T$
(3.27)			$\displaystyle+\left(-\frac{\sigma(1+2\eta)}{2\eta}+\frac{\sigma+\eta}{2\eta}\right)\left(\frac{L_{Q_{10}}(\theta)}{2}+\log T\right)+\frac{\sigma+\eta}{\eta}c_{2}\left(M_{Q_{10}}(\theta)+\log\log T\right).$

•

If $\sigma\leq-\eta$ then as in [HSW22]

(3.28)		$\displaystyle F_{c,r}(\theta)$	$\displaystyle=\log\zeta(1-\sigma)+\frac{1}{2}L_{-1}(\theta)+\left(1+\frac{1-2\sigma}{2}\right)\log T-\frac{1-2\sigma}{2}\log 2\pi$
(3.28)			$\displaystyle+\frac{(1-2\sigma+2[\sigma])}{4}L_{1-[\sigma]}(\theta)+\frac{1}{2}\sum_{j=1}^{-[\sigma]}L_{j-1}(\theta).$

3.4. Conclusion

From (3.4) and (3.5), following [HSW22], if we use Lemma 2.1 we obtain

(3.29)	$\displaystyle\left\|N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$	$\displaystyle\leq\frac{7}{4}+\frac{1}{2}+\frac{1}{25T}+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{1}{\log(r/(c-1/2))}\log\frac{\zeta(c)}{\zeta(2c)}$
		$\displaystyle-\frac{1}{\log(r/(c-1/2))}\log T+\frac{1}{\pi\log(r/(c-1/2))}\int_{0}^{\pi}F_{c,r}(\theta)d\theta$
		$\displaystyle+\frac{E(T,\delta)}{\pi}.$

and hence, being $N_{\mathbb{Q}}(T)=2N(T)$ , we have

(3.30)	$\displaystyle\left\|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$	$\displaystyle\leq\frac{7}{8}+\frac{1}{4}+\frac{1}{50T}+\frac{1}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{1}{2\log(r/(c-1/2))}\log\frac{\zeta(c)}{\zeta(2c)}$
		$\displaystyle-\frac{1}{2\log(r/(c-1/2))}\log T+\frac{1}{2\pi\log(r/(c-1/2))}\int_{0}^{\pi}F_{c,r}(\theta)d\theta$
		$\displaystyle+\frac{E(T,\delta)}{2\pi}.$

If instead we use Lemma 2.3, we have

(3.31)	$\displaystyle\left\|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$	$\displaystyle\leq\frac{7}{8}+\frac{1}{4}+\frac{1}{50T}+\frac{1}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{\log(2.0945\log T)}{2\log(r/(c-1/2))}$
		$\displaystyle-\frac{1}{2\log(r/(c-1/2))}\log T+\frac{1}{2\pi\log(r/(c-1/2))}\int_{0}^{\pi}F_{c,r}(\theta)d\theta$
		$\displaystyle+\frac{E(T,\delta)}{2\pi}.$

As in [Tru14, HSW22], we define

(3.32)

\theta_{y}=\left\{\begin{array}[]{ll}0&\text{ if }c+r\leq y\\ \arccos\frac{y-c}{r}&\text{ if }c-r\leq y\leq c+r;\\ \pi&\text{ if }y\leq c-r.\end{array}\right.

Now, we let $c,r$ , and $\eta$ be positive real numbers satisfying⁴⁴4Note that $\theta_{-1/2}=\pi$ . Indeed, by the definition of $\theta_{y}$ , if we take $y=-1/2$ , then $y\leq c-r$ by the assumption (3.33).

(3.33)

-\frac{1}{2}<c-r<1-c<-\eta<1+\eta<c

and $0<\eta\leq\frac{1}{2}$ . To bound $\int_{0}^{\pi}F_{c,r}(\theta)d\theta$ , we consider the splitting

\int_{0}^{\pi}=\int_{0}^{\theta_{1+\eta}}+\int_{\theta_{1+\eta}}^{\theta_{1}}+\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}+\sum_{h=0}^{n-1}\int_{\sigma_{5+h}}^{\sigma_{4+h}}+\int_{\sigma_{4}}^{\theta_{\frac{1}{2}}}+\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}+\int_{\theta_{0}}^{\theta_{-\eta}}+\int_{\theta_{-\eta}}^{\pi}.

First, we recall two estimates [HSW22] for $L_{j}(\theta)$ defined in (3.23) and $M_{j}(\theta)$ defined in (3.24) which hold for $T\geq T_{0}$ and $\theta\in[0,\pi]$ . Defining, for $\theta\in[0,\pi]$ ,

L_{j}^{\star}(\theta)=\frac{1}{T_{0}}(j+c+r\cos\theta)^{2}+\frac{1}{T_{0}}(r\sin\theta)^{2}+2r\sin\theta,

we have

(3.34)

L_{j}(\theta)\leq\frac{L_{j}^{\star}(\theta)}{T}

and

(3.35)

M_{j}(\theta)\leq\frac{L_{j}^{\star}(\theta)}{2T\log T}.

Now, we proceed with the estimate of each integral as in [HSW22] to derive

(3.36)

\int_{0}^{\theta_{1+\eta}}F_{c,r}(\theta)d\theta\leq\log T\int_{0}^{\theta_{1+\eta}}1d\theta+\int_{0}^{\theta_{1+\eta}}\log\zeta(\sigma)d\theta+\frac{1}{2T}\int_{0}^{\theta_{1+\eta}}L_{-1}^{\star}(\theta)d\theta,

(3.37)			$\displaystyle\int_{\theta_{1+\eta}}^{\theta_{1}}F_{c,r}(\theta)d\theta$
			$\displaystyle\leq\log T\int_{\theta_{1+\eta}}^{\theta_{1}}1d\theta+\frac{c_{2}}{\eta}\log\log T\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)d\theta+\frac{\log c_{1}}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)d\theta$
			$\displaystyle+\frac{\log\zeta(1+\eta)}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(\sigma-1)d\theta+\frac{1}{2T}\int_{\theta_{1+\eta}}^{\theta_{1}}L_{Q_{0}}^{\star}(\theta)d\theta+\frac{c_{2}}{2\eta T\log T}\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)L_{Q_{0}}^{\star}(\theta)d\theta,$

(3.38)			$\displaystyle\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}F_{c,r}(\theta)d\theta$
			$\displaystyle\leq\log T\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}1d\theta+\frac{\log T}{2}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(1-2\sigma+4k_{2}\sigma)d\theta+\log\log T\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(c_{2}(1-2\sigma)+2k_{3}\sigma)d\theta$
			$\displaystyle+\left(\log\frac{c_{1}}{\sqrt{2\pi}}\right)\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(1-2\sigma)d\theta+2\log k_{1}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\sigma d\theta+\frac{1}{2T}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}L_{-1}^{\star}(\theta)d\theta$
			$\displaystyle+\frac{1}{4T}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\left(1-2\sigma+4k_{2}\sigma\right)L_{Q_{11}}^{\star}(\theta)d\theta+\frac{1}{2T\log T}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\left(c_{2}(1-2\sigma)+2k_{3}\sigma\right)L_{Q_{11}}^{\star}(\theta)d\theta,$

(3.39)	$\displaystyle\int_{\theta_{0}}^{\theta_{-\eta}}F_{c,r}(\theta)d\theta$	$\displaystyle\leq\log T\int_{\theta_{0}}^{\theta_{-\eta}}\left(1-\frac{\sigma(1+2\eta)}{2\eta}+\frac{\sigma+\eta}{2\eta}\right)d\theta+\log\log T\int_{\theta_{0}}^{\theta_{-\eta}}\frac{\sigma+\eta}{\eta}c_{2}d\theta$
		$\displaystyle+\int_{\theta_{0}}^{\theta_{-\eta}}\left(-\frac{\sigma}{\eta}\log\frac{1+\eta}{c_{1}(2\pi)^{\eta}}+\log\frac{c_{1}}{\sqrt{2\pi}}\right)d\theta+\frac{1}{2T}\int_{\theta_{0}}^{\theta_{-\eta}}L_{-1}^{\star}(\theta)d\theta$
		$\displaystyle+\frac{1}{T}\int_{\theta_{0}}^{\theta_{-\eta}}\left(-\frac{\sigma(1+2\eta)}{4\eta}+\frac{\sigma+\eta}{4\eta}\right)L_{Q_{10}}^{\star}(\theta)d\theta+\frac{1}{2T\log T}\int_{\theta_{0}}^{\theta_{-\eta}}\frac{\sigma+\eta}{\eta}c_{2}L_{Q_{10}}^{\star}(\theta)d\theta$

and

(3.40)	$\displaystyle\int_{\theta_{-\eta}}^{\pi}F_{c,r}(\theta)d\theta$	$\displaystyle\leq\log T\int_{\theta_{-\eta}}^{\pi}1+\frac{1-2\sigma}{2}d\theta+\int_{\theta_{-\eta}}^{\pi}\log\zeta(1-\sigma)d\theta-\log 2\pi\int_{\theta_{-\eta}}^{\pi}\frac{1-2\sigma}{2}d\theta$
		$\displaystyle+\frac{1}{2T}\int_{\theta_{-\eta}}^{\pi}L_{-1}^{\star}(\theta)d\theta+\frac{1}{T}\int_{\theta_{-\eta}}^{\theta_{-\frac{1}{2}}}\frac{1-2\sigma}{4}L_{1}^{\star}(\theta)d\theta$
		$\displaystyle+\sum_{j=1}^{\infty}\int_{\theta_{-j+\frac{1}{2}}}^{\theta_{-j-\frac{1}{2}}}\left(\frac{1-2\sigma-2j}{4}L_{j+1}(\theta)+\frac{1}{2}\sum_{k=1}^{j}L_{k-1}(\theta)\right)d\theta.$

Observe that

(3.41)			$\displaystyle\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}F_{c,r}(\theta)d\theta$
			$\displaystyle\leq\frac{\log T}{(2^{n+4}-2)(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((2^{n+4}-1)(1-\sigma)+(2^{n+4}-2)(\sigma-\sigma_{n+4})\right)d\theta$
			$\displaystyle+\frac{\log\log T}{1-\sigma_{n+4}}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((1-\sigma)+c_{2}(\sigma-\sigma_{n+4})\right)d\theta$
			$\displaystyle+\frac{\log 1.546}{(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}(1-\sigma)d\theta+\frac{\log c_{1}}{(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}(\sigma-\sigma_{n+4})d\theta$
			$\displaystyle+\frac{1}{2T(2^{n+4}-2)(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((2^{n+4}-1)(1-\sigma)+(2^{n+4}-2)(\sigma-\sigma_{n+4})\right)L_{Q_{0,n+4}}^{\star}(\theta)d\theta$
			$\displaystyle+\frac{1}{2T\log T(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((1-\sigma)+c_{2}(\sigma-\sigma_{n+4})\right)L_{Q_{0,n+4}}^{\star}(\theta)d\theta.$

Hence, for every $0\leq h\leq n-1$ , we have

(3.42)			$\displaystyle\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}F_{c,r}(\theta)d\theta$
			$\displaystyle\leq\log\log T\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}1d\theta+\frac{\log T}{(2^{h+4}-2)(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}$
			$\displaystyle\times\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}\left((2^{h+5}-2)(2^{h+4}-1)(\sigma_{5+h}-\sigma)+(2^{h+4}-2)(2^{h+5}-1)(\sigma-\sigma_{h+4})\right)d\theta$
			$\displaystyle+\log(1.546)\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}1d\theta+\frac{1}{2T\log T}\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}L_{Q_{4+h,5+h}}^{\star}(\theta)d\theta$
			$\displaystyle+\frac{1}{2T(2^{h+4}-2)(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}$
			$\displaystyle\times\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}\left((2^{h+5}-2)(2^{h+4}-1)(\sigma_{5+h}-\sigma)+(2^{h+4}-2)(2^{h+5}-1)(\sigma-\sigma_{h+4})\right)L_{Q_{4+h,5+h}}^{\star}(\theta)d\theta.$

Finally, we have

(3.43)	$\displaystyle\int_{\theta_{\sigma_{4}}}^{\theta_{1/2}}F_{c,r}(\theta)d\theta$	$\displaystyle\leq\frac{\log T}{14(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(14(k_{2}+1)(\sigma_{4}-\sigma)+15\left(\sigma-\frac{1}{2}\right)\right)d\theta$
		$\displaystyle+\frac{\log\log T}{(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(k_{3}(\sigma_{4}-\sigma)+\left(\sigma-\frac{1}{2}\right)\right)d\theta$
		$\displaystyle+\frac{\log k_{1}}{\sigma_{4}-\frac{1}{2}}\int_{\theta_{\sigma_{4}}}^{\theta_{1/2}}(\sigma_{4}-\sigma)d\theta+\frac{\log 1.546}{\sigma_{4}-\frac{1}{2}}\int_{\theta_{\sigma_{4}}}^{\theta_{1/2}}\left(\sigma-\frac{1}{2}\right)d\theta$
		$\displaystyle+\frac{1}{2T14(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(14(k_{2}+1)(\sigma_{4}-\sigma)+15\left(\sigma-\frac{1}{2}\right)\right)L_{Q_{2}}^{\star}(\theta)d\theta$
		$\displaystyle+\frac{1}{2T\log T(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(k_{3}(\sigma_{4}-\sigma)+\left(\sigma-\frac{1}{2}\right)\right)L_{Q_{2}}^{\star}(\theta)d\theta$

With the above estimates in hand, we are ready to estimate the constants.

3.4.1. Constant $C_{1}$

Setting

	$\displaystyle\overline{C}_{1}$	$\displaystyle=\frac{1}{(2^{n+4}-2)(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((2^{n+4}-1)(1-\sigma)+(2^{n+4}-2)(\sigma-\sigma_{n+4})\right)d\theta$
		$\displaystyle+\sum_{h=0}^{n-1}\frac{1}{(2^{h+4}-2)(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}$
		$\displaystyle\times\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}\left((2^{h+5}-2)(2^{h+4}-1)(\sigma_{5+h}-\sigma)+(2^{h+4}-2)(2^{h+5}-1)(\sigma-\sigma_{h+4})\right)d\theta$
		$\displaystyle+\frac{1}{14(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(14(k_{2}+1)(\sigma_{4}-\sigma)+15\left(\sigma-\frac{1}{2}\right)\right)d\theta$
		$\displaystyle+\frac{1}{2}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(1-2\sigma+4k_{2}\sigma)d\theta+\int_{\theta-\eta}^{\pi}\frac{1-2\sigma}{2}d\theta+(\theta_{1}-\theta_{1/2})+\int_{\theta_{0}}^{\theta_{-\eta}}\left(-\frac{\sigma(1+2\eta)}{2\eta}+\frac{\sigma+\eta}{2\eta}\right)d\theta,$

we can express $C_{1}$ as

(3.44)

C_{1}=\frac{\overline{C}_{1}}{2\pi\log(r/(c-1/2))}.

3.4.2. Constants $C_{2}$ and $C^{\prime}_{2}$

Given

	$\displaystyle\overline{C}_{2}$	$\displaystyle=\frac{1}{1-\sigma_{n+4}}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((1-\sigma)+c_{2}(\sigma-\sigma_{n+4})\right)d\theta$
		$\displaystyle+\sum_{h=0}^{n-1}\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}1d\theta+\frac{1}{(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(k_{3}(\sigma_{4}-\sigma)+\left(\sigma-\frac{1}{2}\right)\right)d\theta$
		$\displaystyle+\frac{c_{2}}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)d\theta++\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(c_{2}(1-2\sigma)+2k_{3}\sigma)d\theta+\int_{\theta_{0}}^{\theta_{-\eta}}\frac{\sigma+\eta}{\eta}c_{2}d\theta$
		$\displaystyle=\frac{1}{1-\sigma_{n+4}}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((1-\sigma)+c_{2}(\sigma-\sigma_{n+4})\right)d\theta+\theta_{\sigma_{4}}$
		$\displaystyle-\theta_{\sigma_{n+4}}+\frac{1}{(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(k_{3}(\sigma_{4}-\sigma)+\left(\sigma-\frac{1}{2}\right)\right)d\theta$
		$\displaystyle+\frac{c_{2}}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)d\theta++\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(c_{2}(1-2\sigma)+2k_{3}\sigma)d\theta+\int_{\theta_{0}}^{\theta_{-\eta}}\frac{\sigma+\eta}{\eta}c_{2}d\theta,$

by Lemma 2.1, our $C_{2}$ can be written as

(3.45)

C_{2}=\frac{\overline{C}_{2}}{2\pi\log(r/(c-1/2))};

while Lemma 2.3 implies that

(3.46)

C^{\prime}_{2}=\frac{\overline{C}_{2}}{2\pi\log(r/(c-1/2))}+\frac{1.2322}{2\log(r/(c-1/2))}.

3.4.3. Constants $C_{3}$ and $C^{\prime}_{3}$

Using Lemma 2.1, $C_{3}$ can be expressed by

(3.47)		$\displaystyle C_{3}$	$\displaystyle=\frac{7}{8}+\frac{1}{4}+\frac{1}{50T}+\frac{1}{\pi}\log\zeta(\sigma_{1})+\frac{1}{2\log(r/(c-1/2))}\log\frac{\zeta(c)}{\zeta(2c)}+\frac{1}{2}\left(\frac{640\delta-112}{1536\left(3T_{0}-1\right)}+\frac{1}{2^{10}}\right)$
(3.47)			$\displaystyle+\frac{1}{2\pi\log(r/(c-1/2))}\left(D_{3}+\kappa_{1}\left(J_{1}\right)+\kappa_{2}\left(J_{2}\right)+\kappa_{3}\left(T_{0}\right)\right),$

while Lemma 2.3 implies that the constant $C^{\prime}_{3}$ will be

(3.48)		$\displaystyle C^{\prime}_{3}$	$\displaystyle=\frac{7}{8}+\frac{1}{4}+\frac{1}{50T}+\frac{1}{\pi}\log\zeta(\sigma_{1})+\frac{1}{2}\left(\frac{640\delta-112}{1536\left(3T_{0}-1\right)}+\frac{1}{2^{10}}\right)$
(3.48)			$\displaystyle+\frac{1}{2\pi\log(r/(c-1/2))}\left(D_{3}+\kappa_{1}\left(J_{1}\right)+\kappa_{2}\left(J_{2}\right)+\kappa_{3}\left(T_{0}\right)\right),$

where

	$\displaystyle D_{3}$	$\displaystyle=\frac{\log c_{1}}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(1+\eta-\sigma)d\theta+\frac{\log\zeta(1+\eta)}{\eta}\int_{\theta_{1+\eta}}^{\theta_{1}}(\sigma-1)d\theta$
		$\displaystyle+\frac{\log 1.546}{(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}(1-\sigma)d\theta+\frac{\log c_{1}}{(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}(\sigma-\sigma_{n+4})d\theta$
		$\displaystyle+\sum_{h=0}^{n-1}\log(1.546)\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}d\theta+\frac{\log k_{1}}{\sigma_{4}-\frac{1}{2}}\int_{\theta_{\sigma_{4}}}^{\theta_{1/2}}(\sigma_{4}-\sigma)d\theta+\frac{\log 1.546}{\sigma_{4}-\frac{1}{2}}\int_{\theta_{\sigma_{4}}}^{\theta_{1/2}}\left(\sigma-\frac{1}{2}\right)d\theta$
		$\displaystyle+\left(\log\frac{c_{1}}{\sqrt{2\pi}}\right)\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}(1-2\sigma)d\theta+2\log k_{1}\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\sigma d\theta$
		$\displaystyle+\int_{\theta_{0}}^{\theta_{-\eta}}\left(-\frac{\sigma}{\eta}\log\frac{1+\eta}{c_{1}(2\pi)^{\eta}}+\log\frac{c_{1}}{\sqrt{2\pi}}\right)d\theta-(\log 2\pi)\int_{\theta_{-\eta}}^{\pi}\frac{1-2\sigma}{2}d\theta$
		$\displaystyle+\frac{\log\zeta(1+\eta)+\log\zeta(c)}{2}\left(\theta_{1+\eta}-\frac{\pi}{2}\right)+\frac{\pi}{4J_{1}}\log\zeta(c)$
		$\displaystyle+\frac{\log\zeta(1+\eta)+\log\zeta(c)}{2}\left(\theta_{1-c}-\theta_{-\eta}\right)+\frac{\pi-\theta_{1-c}}{2J_{2}}\log\zeta(c),$

\kappa_{1}\left(J_{1}\right)=\frac{\pi}{4J_{1}}\left(\log\zeta(c+r)+2\sum_{j=1}^{J_{1}-1}\log\zeta\left(c+r\cos\frac{\pi j}{2J_{1}}\right)\right),

(3.49)

\displaystyle\kappa_{2}\left(J_{2}\right)=

\displaystyle\frac{\pi-\theta_{1-c}}{2J_{2}}(\log\zeta(1-c+r)+2\sum_{j=1}^{J_{2}-1}\log\zeta\left(1-c-r\cos\left(\frac{\pi j}{J_{2}}+\left(1-\frac{j}{J_{2}}\right)\theta_{1-c}\right)\right),

and

\displaystyle\kappa_{3}(T_{0})=\frac{1}{2T_{0}}\max\left\{0,\mathscr{M}_{1}\right\}+\frac{1}{2T_{0}\log\log T_{0}}\max\left\{0,\mathscr{M}_{2}\right\},

with

	$\displaystyle\mathscr{M}_{1}=\int_{0}^{\theta_{1+\eta}}L_{-1}^{\star}(\theta)+\int_{\theta_{1+\eta}}^{\theta_{1}}L_{Q_{0}}^{\star}(\theta)d\theta+\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\left(L_{-1}^{\star}(\theta)+\frac{1}{2}\left(1-2\sigma+4k_{2}\sigma\right)L_{Q_{11}}^{\star}(\theta)\right)d\theta$
	$\displaystyle+\frac{1}{(2^{n+4}-2)(1-\sigma_{n+4})}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((2^{n+4}-1)(1-\sigma)+(2^{n+4}-2)(\sigma-\sigma_{n+4})\right)L_{Q_{0,n+4}}^{\star}(\theta)d\theta$
	$\displaystyle+\sum_{h=0}^{n-1}\frac{1}{(2^{h+4}-2)(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}$
	$\displaystyle\times\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}\left((2^{h+5}-2)(2^{h+4}-1)(\sigma_{5+h}-\sigma)+(2^{h+4}-2)(2^{h+5}-1)(\sigma-\sigma_{h+4})\right)L_{Q_{4+h,5+h}}^{\star}(\theta)d\theta$
	$\displaystyle+\frac{1}{14(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(14(k_{2}+1)(\sigma_{4}-\sigma)+15\left(\sigma-\frac{1}{2}\right)\right)L_{Q_{2}}^{\star}(\theta)d\theta$
	$\displaystyle+\int_{\theta_{0}}^{\theta_{-\eta}}\left(L_{-1}^{\star}(\theta)+\left(-\sigma+\frac{1}{2}\right)L_{Q_{10}}^{\star}(\theta)\right)d\theta+\int_{\theta_{-\eta}}^{\pi}\left(L_{-1}^{\star}(\theta)+\frac{1-2\sigma}{2}L_{1}^{\star}(\theta)\right)d\theta$

and

	$\displaystyle\mathscr{M}_{2}=\int_{\theta_{1+\eta}}^{\theta_{1}}\frac{c_{2}}{\eta}(1+\eta-\sigma)L_{Q_{0}}^{\star}(\theta)d\theta+\frac{1}{1-\sigma_{n+4}}\int_{\theta_{1}}^{\theta_{\sigma_{n+4}}}\left((1-\sigma)+c_{2}(\sigma-\sigma_{n+4})\right)L_{Q_{0,n+4}}^{\star}(\theta)d\theta$
	$\displaystyle+\sum_{h=0}^{n-1}\int_{\theta_{\sigma_{5+h}}}^{\theta_{\sigma_{4+h}}}L_{Q_{4+h,5+h}}^{\star}(\theta)d\theta+\frac{1}{(\sigma_{4}-\frac{1}{2})}\int_{\theta_{\sigma_{4}}}^{\theta_{{1/2}}}\left(k_{3}(\sigma_{4}-\sigma)+\left(\sigma-\frac{1}{2}\right)\right)L_{Q_{2}}^{\star}(\theta)d\theta$
	$\displaystyle+\int_{\theta_{\frac{1}{2}}}^{\theta_{0}}\left(c_{2}(1-2\sigma)+2k_{3}\sigma\right)L_{Q_{11}}^{\star}(\theta)d\theta+\int_{\theta_{0}}^{\theta_{-\eta}}\frac{\sigma+\eta}{\eta}c_{2}L_{Q_{10}}^{\star}(\theta)d\theta.$

The proof of Theorem 1.2 is complete.

4. Proof of Theorem 1.4

Let $T\geq T_{0}$ fixed. We recall that

(4.1)

S(T)=\frac{1}{\pi}\Delta_{\mathscr{C}_{0}}\arg\zeta(s)=\frac{1}{\pi}\Delta_{\mathscr{C}_{1}}\arg\zeta(s)+\frac{1}{\pi}\Delta_{\mathscr{C}_{2}}\arg(s-1)\zeta(s)-\frac{1}{\pi}\Delta_{\mathscr{C}_{2}}\arg(s-1),

where

(4.2)

\left|\Delta_{\mathscr{C}_{1}}\arg\zeta(s)\right|\leq\log\zeta\left(\sigma_{1}\right)

and

(4.3)

\left|\Delta_{\mathscr{C}_{2}}\arg(s-1)\right|=\arctan\left(\frac{\sigma_{1}-1}{T}\right)+\arctan\left(\frac{1}{2T}\right)\leq\arctan\left(\frac{\sigma_{1}-1}{T_{0}}\right)+\arctan\left(\frac{1}{2T_{0}}\right)

for $T\geq T_{0}$ . By (3.4) and Theorem 1.2,

(4.4)		$\displaystyle\left\|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$	$\displaystyle\leq\frac{7}{8}+\frac{1}{50T}+\frac{1}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{1}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
(4.4)			$\displaystyle\leq C_{1}\log T+\min\{C_{2}\log\log T+C_{3},C^{\prime}_{2}\log\log T+C^{\prime}_{3}\}$

and hence

	$\displaystyle\frac{1}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
	$\displaystyle\leq C_{1}\log T+\min\{C_{2}\log\log T+C_{3},C^{\prime}_{2}\log\log T+C^{\prime}_{3}\}-\frac{7}{8}-\frac{1}{50T}-\frac{1}{\pi}\log\zeta\left(\sigma_{1}\right).$

It follows that, for $T\geq T_{0}$ , the following estimate holds:

(4.5)

|S(T)|\leq C_{1}\log T+\min\{C_{2}\log\log T+\tilde{C_{3}},C^{\prime}_{2}\log\log T+\tilde{C^{\prime}_{3}}\},

where

(4.6)

\tilde{C_{3}}=C_{3}-\frac{7}{8}-\frac{1}{50T}+\frac{1}{\pi}\left(\arctan\left(\frac{\sigma_{1}-1}{T_{0}}\right)+\arctan\left(\frac{1}{2T_{0}}\right)\right)

and

(4.7)

\tilde{C^{\prime}_{3}}=C^{\prime}_{3}-\frac{7}{8}-\frac{1}{50T}+\frac{1}{\pi}\left(\arctan\left(\frac{\sigma_{1}-1}{T_{0}}\right)+\arctan\left(\frac{1}{2T_{0}}\right)\right).

This completes the proof of Theorem 1.4.

5. Proofs of Theorem 1.1 and Corollary 1.5

First, we apply Theorem 1.2 and Theorem 1.4 with $T_{0}=30610046000$ . Furthermore, we take $J_{1}=64$ , $J_{2}=39$ and we choose the following parameters $Q_{i}$ :

(Q_{0},Q_{1},Q_{2},Q_{3},Q_{4},Q_{5},Q_{6},Q_{7},Q_{8},Q_{9},Q_{10},Q_{11})=(1,1.18,1.18,3.9,1,1,1,1,1,1,2.3,3.9).

The first row of Table 2 gives Theorem 1.1 and Corollary 1.5 for $T\geq 30610046000$ .

$c$	$r$	$\eta$	$C_{1}$	$C_{2}$	$C_{2}^{\prime}$	$C_{3}$	$C_{3}^{\prime}$	$\tilde{C_{3}}$	$\tilde{C^{\prime}_{3}}$
$1.000225$	$1.000605$	$0.000158$	$0.10076$	$0.24460$	$1.13325$	$8.08292$	$2.38404$	$7.20792$	$1.50904$
$1.000060$	$1.499556$	$1.542440\cdot 10^{-5}$	$0.12355$	$0.06782$	$0.62883$	$6.25781$	$2.05840$	$5.38281$	$1.18340$
$1.499159$	$1.998357$	$0.499050$	$0.16732$	$0.17266$	$1.06148$	$1.96275$	$1.40212$	$1.08775$	$0.52712$
$1.043400$	$1.250450$	$0.040000$	$0.11200$	$0.12567$	$0.86492$	$3.77389$	$2.14756$	$2.89889$	$1.27256$

Table 2. Some admissible values for

C_{1},C_{2},C^{\prime}_{2},C_{3},C^{\prime}_{3},

\tilde{C_{3}}

\tilde{C^{\prime}_{3}}

For $e\leq T\leq 30610046000$ , in order to estimate $|S(T)|$ , we use the known bound (1.2) computed by Platt, which is in particular sharper than

0.10076\log T+\min\{0.24460\log\log T+7.20792,1.13325\log\log T+1.50904\}

for every $e\leq T\leq 30610046000$ . Hence, Corollary 1.5 follows by combining the two cases.
Now, it remains to prove Theorem 1.1 when $e\leq T\leq 30610046000$ . As per [HSW22], since

(5.1)

N(T)=S(T)+\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)+\frac{7}{8}+\frac{g(T)}{2},

for $e\leq T\leq 30610046000$ we have

(5.2)

\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq|S(T)|+\frac{1}{2}|g(T)|+\frac{7}{8}\leq 2.5167+\frac{1}{50e}+\frac{7}{8},

which is always smaller than

0.10076\log T+0.24460\log\log T+8.08292

for every $e\leq T\leq 30610046000$ . Hence, combining the two cases, Theorem 1.1 follows.

6. Proof of Corollary 1.6

Since

(6.1)

N(T)=S(T)+\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)+\frac{7}{8}+\frac{g(T)}{2},

one has

(6.2)

\displaystyle N(T+1)-N(T)=S(T+1)-S(T)+\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)+\frac{g(T+1)-g(T)}{2}

and

(6.3)			$\displaystyle N(T+1)-N(T-1)$
(6.3)			$\displaystyle=S(T+1)-S(T-1)+\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T-1}{2\pi}\log\left(\frac{T-1}{2\pi e}\right)+\frac{g(T+1)-g(T-1)}{2}.$

Writing

\log\left(\frac{T+1}{2\pi e}\right)=\log\left(\frac{T}{2\pi e}\right)+\log\left(1+\frac{1}{T}\right)\quad\text{and}\quad\log\left(\frac{T-1}{2\pi e}\right)=\log\left(\frac{T}{2\pi e}\right)+\log\left(1-\frac{1}{T}\right),

and using the Taylor expansions at $x=0$ and $y=\infty$

\displaystyle\log\left(1+x\right)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+O(x^{5})\quad\text{and}\quad\log\left(\frac{1+y}{-1+y}\right)=\frac{2}{y}+\frac{2}{3y^{3}}+\frac{2}{5y^{5}}+O\left(\frac{1}{y^{6}}\right)

with $x=1/T$ , $x=-1/T$ , $y=T$ , we have, for $T\geq 2$ ,

(6.4)			$\displaystyle\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)$
			$\displaystyle=\frac{T}{2\pi}\log\left(1+\frac{1}{T}\right)+\frac{1}{2\pi}\left(\log\left(\frac{T}{2\pi e}\right)+\log\left(1+\frac{1}{T}\right)\right)$
			$\displaystyle=\frac{1}{2\pi}\left(1-\frac{1}{2T}+\frac{1}{3T^{2}}-\cdots\right)+\frac{1}{2\pi}\left(\frac{1}{T}-\frac{1}{2T^{2}}+\frac{1}{3T^{3}}-\cdots\right)+\frac{1}{2\pi}\log T-\frac{1}{2\pi}\log(2\pi e)$
			$\displaystyle<\frac{1}{2\pi}\log T+\frac{3}{4\pi}-\frac{1}{2\pi}\log(2\pi e),$

and, similarly,

(6.5)			$\displaystyle\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T-1}{2\pi}\log\left(\frac{T-1}{2\pi e}\right)$
			$\displaystyle=\frac{T}{2\pi}\left(\log\left(1+\frac{1}{T}\right)-\log\left(1-\frac{1}{T}\right)\right)+\frac{1}{2\pi}\left(2\log\left(\frac{T}{2\pi e}\right)+\log\left(1+\frac{1}{T}\right)+\log\left(1-\frac{1}{T}\right)\right)$
			$\displaystyle<\frac{1}{\pi}\log T+\frac{\log 3}{\pi}-\frac{1}{\pi}\log(2\pi e).$

Furthermore,

(6.6)		$\displaystyle\left\|\frac{g(T+1)-g(T)}{2}\right\|$	$\displaystyle\leq\max\{\|g(T+1)\|,\|g(T)\|\}\leq\frac{1}{25T},$
(6.6)		$\displaystyle\left\|\frac{g(T+1)-g(T-1)}{2}\right\|$	$\displaystyle\leq\max\{\|g(T+1)\|,\|g(T-1)\|\}\leq\frac{1}{25(T-1)}.$

Finally, if $T+1>30610046000$ , then Theorem 1.4 implies that

(6.7)

\displaystyle|S(T+1)-S(T)|\leq 2.00001C_{1}\log T+2.00001\min\{C_{2}\log\log T+\tilde{C_{3}},C^{\prime}_{2}\log\log T+\tilde{C^{\prime}_{3}}\}

and

(6.8)

\displaystyle|S(T+1)-S(T-1)|\leq 2.00001C_{1}\log T+2.00001\min\{C_{2}\log\log T+\tilde{C_{3}},C^{\prime}_{2}\log\log T+\tilde{C^{\prime}_{3}}\},

while if $3\leq T+1\leq 30610046000$ , by (1.2) we get

(6.9)

\displaystyle|S(T+1)-S(T)|\leq 2\cdot 2.5167\leq 5.0334

and

(6.10)

\displaystyle|S(T+1)-S(T-1)|\leq 2\cdot 2.5167\leq 5.0334.

Substituting the estimates we found above into (6.2) and (6.3), we can conclude that, for $T+1>30610046000$ , one has

(6.11)			$\displaystyle N(T+1)-N(T)$
(6.11)			$\displaystyle\leq\left(\frac{1}{2\pi}+2C_{1}\right)\log T+2\min\{C_{2}\log\log T+\tilde{C_{3}},C^{\prime}_{2}\log\log T+\tilde{C^{\prime}_{3}}\}+\frac{3}{4\pi}-\frac{1}{2\pi}\log(2\pi e)+\frac{1}{25T},$

and

		$\displaystyle N(T+1)-N(T-1)$
		$\displaystyle\leq\left(\frac{1}{\pi}+2C_{1}\right)\log T+\min\{2C_{2}\log\log T+\mathscr{D}_{3},2C^{\prime}_{2}\log\log T+\mathscr{D^{\prime}}_{3}\}+\frac{1}{25(T-1)},$

where

\mathscr{D}_{3}=2\tilde{C_{3}}+\frac{\log 3-\log(2\pi e)}{\pi}\quad\text{and}\quad\mathscr{D^{\prime}}_{3}=2\tilde{C^{\prime}_{3}}+\frac{\log 3-\log(2\pi e)}{\pi}.

On the other hand, if $3\leq T+1\leq 30610046000$ then

(6.12)

N(T+1)-N(T)\leq\frac{1}{2\pi}\log T+5.0334+\frac{3}{4\pi}-\frac{1}{2\pi}\log(2\pi e)+\frac{1}{25T}\leq\frac{1}{2\pi}\log T+4.8405

and

(6.13)

N(T+1)-N(T-1)\leq\frac{1}{\pi}\log T+5.0334+\frac{\log 3}{\pi}-\frac{1}{\pi}\log(2\pi e)\leq\frac{1}{\pi}\log T+4.4798.

Finally, it remains to find the lower bounds. Similarly to what we did in (6.4), (6.5),

(6.14)

\displaystyle\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)>\frac{1}{2\pi}\log T+\frac{1}{2\pi}-\frac{1}{2\pi}\log(2\pi e),

and

(6.15)

\displaystyle\frac{T+1}{2\pi}\log\left(\frac{T+1}{2\pi e}\right)-\frac{T-1}{2\pi}\log\left(\frac{T-1}{2\pi e}\right)>\frac{1}{\pi}\log T+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e).

It follows that, for $T+1>30610046000$ ,

(6.16)			$\displaystyle N(T+1)-N(T-1)$
			$\displaystyle>\frac{1}{\pi}\log T+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e)-\|S(T+1)-S(T-1)\|-\frac{1}{25(T-1)}$
			$\displaystyle>\left(\frac{1}{\pi}-2.000001C_{1}\right)\log T-2\min\{C_{2}\log\log T+\mathscr{E},C^{\prime}_{2}\log\log T+\mathscr{E^{\prime}}\}-\frac{1}{25(T-1)},$

where

\mathscr{E}=2\tilde{C_{3}}+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e)\quad\text{and}\quad\mathscr{E^{\prime}}=2\tilde{C^{\prime}_{3}}+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e).

On the other hand, if $3\leq T+1\leq 30610046000$ , then

N(T+1)-N(T)>\frac{1}{2\pi}\log T+\frac{1}{2\pi}-\frac{1}{2\pi}\log(2\pi e)-5.0334-\frac{1}{25T}

and

N(T+1)-N(T-1)>\frac{1}{\pi}\log T+\frac{1}{\pi}+\frac{\log(3/4)}{2\pi}-\frac{1}{\pi}\log(2\pi e)-5.0334-\frac{1}{25(T-1)}.

Remark.

The $0.000001$ goes inside the approximation of the last decimal place in the various constants.

Acknowledgments

The authors thank Andrew Fiori, Nathan Ng, David Platt and Tim Trudgian for the helpful discussions and comments. The first author also thanks the organizers of the workshop “Analytic and Explicit results of zeros of $L$ -functions” (Bȩdlewo, September 23-27, 2024), where helpful discussions with David Platt took place.

References

[Lmf] The LMFDB Collaboration, The $L$ -functions and modular forms database URL: https://www.lmfdb.org/
[Bac16] R. J. Backlund “Über die Nullstellen der Riemannschen Zetafunktion” In Acta Math. 41.1, 1916, pp. 345–375 DOI: 10.1007/BF02422950
[Bel] Chiara Bellotti “Argument and zero counting function estimates for zeta” GitHub repository available at URL: https://github.com/ChiaraBellotti/Argument-and-counting-function-estimates-for-Riemann-zeta-function
[Bel24] Chiara Bellotti “Explicit bounds for the Riemann zeta function and a new zero-free region” In J. Math. Anal. Appl. 536.2, 2024, pp. 128249
[Ben+21] Michael A. Bennett, Greg Martin, Kevin O’Bryant and Andrew Rechnitzer “Counting zeros of Dirichlet $L$ -functions” In Math. Comp. 90.329, 2021, pp. 1455–1482
[Dud16] Adrian W. Dudek “An explicit result for primes between cubes” In Funct. Approx. Comment. Math. 55.2, 2016, pp. 177 –197
[FK15] L. Faber and H. Kadiri “New bounds for $\psi(x)$ ” In Math. Comp. 84.293, 2015, pp. 1339–1357 DOI: 10.1090/S0025-5718-2014-02886-X
[FGH05] David Farmer, S. Gonek and Christopher Hughes “The maximum size of $L$ -functions” In Journal für die reine und angewandte Mathematik (Crelles Journal) 2007, 2005 DOI: 10.1515/CRELLE.2007.064
[Gro13] J. Grossmann “Über die Nullstellen der Riemannschen Zeta-Funktion und der Dirichletschen $L$ -Funktionen” PhD thesis, Georg-August-Universität Göttingen, 1913
[HSW21] Elchin Hasanalizade, Quanli Shen and PENG-JIE Wong “Counting zeros of Dedekind zeta functions” In Math. Comp. 91, 2021, pp. 277–293
[HSW22] Elchin Hasanalizade, Quanli Shen and Peng-Jie Wong “Counting zeros of the Riemann zeta function” In J. Number Theory 235, 2022, pp. 219–241
[Hia16] Ghaith A. Hiary “An explicit van der Corput estimate for $\zeta(1/2+it)$ ” In Indag. Math. 27.2, 2016, pp. 524–533
[HLY24] Ghaith A. Hiary, Nicol Leong and Andrew Yang “Explicit bounds for the Riemann zeta-function on the 1-line” (to appear) In Funct. Approx. Comment. Math., 2024
[HPY24] Ghaith A. Hiary, Dhir Patel and Andrew Yang “An improved explicit estimate for $\zeta(1/2+it)$ ” In J. Number Theory 256, 2024, pp. 195–217
[JCH24] Daniel R. Johnston and Michaela Cully-Hugill “On the error term in the explicit formula of Riemann-von Mangoldt II” ArXiv:2402.04272, 2024
[Leo24] Nicol Leong “Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta-function” ArXiv:2405.04869, 2024
[Pat22] Dhir Patel “An explicit upper bound for $\zeta(1+it)$ ” In Indag. Math. 33.5, 2022, pp. 1012–1032
[PY24] Dhir Patel and Andrew Yang “An explicit sub-Weyl bound for $\zeta(1/2+it)$ ” To appear In J. Number Theory, 2024
[Pin87] J. Pintz “An effective disproof of the Mertens conjecture” In Journées arithmétiques de Besançon, Astérisque 147-148 Société mathématique de France, 1987, pp. 325–333
[PT21] Dave Platt and Tim Trudgian “The Riemann hypothesis is true up to $3\cdot 10^{12}$ ” In Bull. Lond. Math. Soc. 53.3, 2021, pp. 792–797 DOI: https://doi.org/10.1112/blms.12460
[Pla16] David Platt “Isolating some non-trivial zeros of zeta” In Math. Comp. 86, 2016, pp. 1 DOI: 10.1090/mcom/3198
[PT15] D.J. Platt and T.S. Trudgian “An improved explicit bound on $|\zeta(1/2+it)|$ ” In J. Number Theory 147, 2015, pp. 842–851
[QT24] Eunice Hoo Qingyi and Lee-Peng Teo “Explicit Bound of $|\zeta\left(1+it\right)|$ ” ArXiv:2412.00766, 2024
[Ros41] J. B. Rosser “Explicit bounds for some functions of prime numbers” In Amer. J. Math. 63, 1941, pp. 211–232 DOI: 10.2307/2371291
[RK23] John Rozmarynowycz and Seungki Kim “A new upper bound on the smallest counterexample to the Mertens conjecture” ArXiv:2305.00345, 2023
[Tru14] T. S. Trudgian “An improved upper bound for the argument of the Riemann zeta-function on the critical line II” In J. Number Theory 134, 2014, pp. 280–292 DOI: 10.1016/j.jnt.2013.07.017
[Tru12] Timothy Trudgian “An improved upper bound for the argument of the riemann zeta-function on the critical line” In Math. Comp. 81.278, 2012, pp. 1053 – 1061
[VM05] H. C. F. Von Mangoldt “Zur Verteilung der Nullstellen der Riemannschen Funktion $\xi(t)$ ” In Math. Ann. 60.1, 1905, pp. 1–19 DOI: 10.1007/BF01447494
[Yan24] Andrew Yang “Explicit bounds on $\zeta(s)$ in the critical strip and a zero-free region” In J. Math. Anal. Appl. 534.2, 2024, pp. 128124

Appendix A A Numerical Study

Andrew Fiori ⁵⁵5 Department of Mathematics and Statistics, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4, Canada
Andrew Fiori’s Research is supported by NSERC Discovery Grant RGPIN-2020-05316.

Under the Riemann Hypothesis one expects a bound of the form:

(A.1)

\left|N(T)-\Big{(}\frac{T}{2\pi}\log\frac{T}{2\pi e}+7/8\Big{)}\right|=C{\log(T)\log\log(T)}+O(1).

However, it is conjectured in [FGH05] that the actual size is:

(A.2)

\left|N(T)-\Big{(}\frac{T}{2\pi}\log\frac{T}{2\pi e}+7/8\Big{)}\right|<\frac{1}{\sqrt{2}\pi}\sqrt{\log(T)\log\log(T)}+o(1).

One goal of this appendix is to provide numerical evidence for this stronger conjecture. A second goal is to provide useful input for future works to improve upon effective bounds on $N(T)$ by summarizing what is known using the work of [Pla16]. We provide such a result in Theorem A.4.

Here we study $T<30\,610\,046\,000$ by using the list of the zeros for the zeta function as computed by [Pla16] and made available at [Lmf]. This database of the first $103\,800\,788\,359$ many zeros is broken up into 14 580 intervals. For practical reasons we analyze the data using these intervals. We note that although [PT21] has verified RH to $3\cdot 10^{12}$ , they have not produced a database of zeros.

We shall denote the imaginary part of the $n$ th zero, ordered by height $t_{n}$ . Because all the zeros up to $30\,610\,046\,000$ are simple we know that in the interval under consideration $N(t_{n})=n$ .

A.1. Average Values

Observation A.1.

On each of the intervals of zeros produced by [Pla16], the average value of the function

N(t_{n})-\frac{t_{n}}{2\pi}\log\frac{t_{n}}{2\pi e}

evaluated at the zeros, $t_{n}$ , of zeta in that interval is approximately $11/8$ .
Excluding the first two intervals, where the deviation from the average is respectively $-5.07685\cdot 10^{-05}$ and $2.48229\cdot 10^{-05}$ , on each of the intervals the deviation of these average values from $11/8$ is bounded by $5.6678910^{-07}$ .

A.2. Range of Values

Proposition A.2.

On any interval beginning and ending at zeros of $\zeta$ the maximum value of

\epsilon^{+}(T)=N(T)-\Big{(}\frac{T}{2\pi}\log\frac{T}{2\pi e}+11/8\Big{)}-\frac{1}{\sqrt{2}\pi}\sqrt{\log(T)\log\log(T)}

will be taken at $t_{n}$ , the exact ordinate of a zero of $\zeta$ . Moreover, if all zeros on the interval are simple, then the infimum of

\epsilon^{-}(T)=N(T)-\Big{(}\frac{T}{2\pi}\log\frac{T}{2\pi e}+11/8\Big{)}+\frac{1}{\sqrt{2}\pi}\sqrt{\log(T)\log\log(T)}

will be exactly $1$ less than the value $\epsilon^{-}(T)$ takes on at $t_{n}$ , the exact ordinate of a zero of $\zeta$ .

Proof.

Notice that the respective functions $\epsilon^{\pm}(T)$ are continuous and decreasing on any interval $[t_{n},t_{n+1})$ between consecutive zeros and that

\lim_{t\rightarrow t_{n+1}^{-}}\epsilon^{-}(t)=\epsilon^{-}(t_{n+1})-1.

The results follow immediately. ∎

As a result of the above, we focus our attention on the study of minimum values of $\epsilon^{-}(t_{n})$ and maximum values of $\epsilon^{+}(t_{n})$ at zeta zeros.

Observation A.3.

The maximum value of $\epsilon^{+}(t_{n})$ in the database of zeros produced by [Pla16] is approximately $0.0920937$ and occurs at $n=2953463649$ with $t_{n}$ approximately $1035537870.14791389$ . The minimum value of $\epsilon^{-}(t_{n})$ in the database of zeros produced by [Pla16] is approximately $-0.0827069$ and occurs at $n=48227304665$ with $t_{n}$ approximately $14727556977.25899340$ .

Theorem A.4.

For $e<T<30\,610\,046\,000$ we have

-\frac{\sqrt{\log(T)\log\log(T)}}{\sqrt{2}\pi}-1-0.082707<N(T)-\frac{T}{2\pi}\log\frac{T}{2\pi e}-\frac{11}{8}<\frac{\sqrt{\log(T)\log\log(T)}}{\sqrt{2}\pi}+0.092094.

Proof.

This is an immediate consequence of the previous observation and proposition. ∎

A.3. Extreme Values are Rare

The following observation says that in some sense extreme values are rare.

Observation A.5.

The function $\epsilon^{+}(t_{n})$ is almost always negative, indeed, it is negative at all zeros with $0<t<30,610,046,000$ except for those listed in Table 3. Similarly, the function $\epsilon^{-}(t_{n})$ is almost always positive, indeed, it is positive at all zeros with $0<t<30,610,046,000$ for those listed in Table 3. We notice that the frequency of the extreme values is decreasing as both $n$ and $t_{n}$ increase. We also note that there are no examples of consecutive extreme values.

Table 3. List of all positive (respectively negative) values of

\epsilon^{+}(t_{n})

(respectively

\epsilon^{-}(t_{n})

) for

0<t<30,610,046,000

$n$	$t_{n}$	$\epsilon^{+}(t_{n})$
7330779	3745331.534911	0.0045727
10014001	4998855.443421	0.0228842
30930930	14253736.600191	0.0215612
106941331	45420475.080263	0.0548687
121934174	51361501.783167	0.0633788
342331986	135399343.427052	0.0852310
486250460	188404036.065583	0.0077204
1333195695	487931556.151002	0.0711065
1819794287	654800601.959837	0.0016382
2953463649	1035537870.147914	0.0920937
4711070126	1611978781.026883	0.0552043
6020412879	2034221491.431262	0.0040533
6276413932	2116223525.742432	0.0136917
6916958115	2320709265.610272	0.0183240
7895552868	2631384288.230762	0.0134043
18019870103	5765666759.059866	0.0101465
29425625937	9196418366.325099	0.0141895
31587712923	9839079152.616086	0.0169750
43668302178	13396993184.932842	0.0314554
44121363503	13529486654.222228	0.0049450
71876944166	21550885080.446041	0.0076388
100093914039	29565113205.570534	0.0184384

$n$	$t_{n}$	$\epsilon^{-}(t_{n})$
337917	223936.368134	-0.0206077
2009961	1137116.070608	-0.0268423
10869861	5393528.443012	-0.0021561
13999527	6820051.890986	-0.0219395
37592217	17095484.271828	-0.0359387
83088045	35862210.311523	-0.0463096
88600097	38084045.549954	-0.0491801
141617808	59096901.323297	-0.0082036
164689303	68084444.336913	-0.0322461
191297537	78359876.488247	-0.0148321
225291159	91369499.494965	-0.0092099
566415149	217536164.326180	-0.0121163
1081300142	400354486.072002	-0.0593335
1257893678	461849910.598599	-0.0262264
1372703319	501584522.950737	-0.0002137
1955876862	701027396.312615	-0.0096394
2305634166	819113670.556185	-0.0026561
5134032906	1748936581.577121	-0.0142280
5136505385	1749735519.272913	-0.0508501
8864769308	2937266043.546390	-0.0418297
9430966584	3115208316.829027	-0.0490178
9532704476	3147127461.727906	-0.0055879
18629248201	5951053644.636571	-0.0008106
19859326408	6324431638.934122	-0.0452773
21082098810	6694540279.310641	-0.0362361
22909699222	7245905144.854708	-0.0030872
48227304665	14727556977.258993	-0.0827069
77728515578	23222574401.823281	-0.0515078
86585440777	25742609309.393488	-0.0073926
87198634344	25916653877.755976	-0.0017881
97495263831	28831591819.434777	-0.0222335
103274388030	30461757456.864450	-0.0421276

Conjecture A.6.

One could make a variety of conjectures of different strengths based on Table 3.

(1)

The number of extreme values up to height $T$ is less than $C\log T$ for some constant $C$ .
(2)

The set of $n$ for which these functions are respectively positive or negative has natural density zero.

A.4. Extreme Values are Common

The following observation says that in some sense extreme values are common.

Observation A.7.

If for each interval of zeros produced by [Pla16], we compute the maximum value of $\epsilon^{+}(t_{n})$ on that interval, then the minimum of these values is $-0.362463$ . Similarly, if for each interval of zeros produced by [Pla16] we compute the minimum value of $\epsilon^{-}(t_{n})$ on that interval, then the maximum of these values is $0.361383$ .
For context note that $\frac{1}{\sqrt{2}\pi}\sqrt{\log(3\cdot 10^{10})\log\log(3\cdot 10^{10})}$ is approximately $1.97241$ and $2\pi/\log(3\cdot 10^{10}/(2\pi e))$ is approximately $0.295171$ . In particular, for each interval the function

N(t_{n})-\frac{t_{n}}{2\pi}\log\frac{t_{n}}{2\pi e}-11/8

takes on values relatively close to both the theoretical upper and lower extremes under consideration.

A.5. Clusters of Zeros

One common use of estimates on $N(t)$ is to bound the number of zeros on an interval. The quality of the bounds one obtains using bounds on $N(t)$ improves with $t$ . Consequently, it is useful to have explicit bounds on the number of zeros in intervals for small $t$ .

Observation A.8.

For $e<t<30\;610\;045\;999$ , the maximum value for $\frac{N(t+1)-N(t-1)}{\log t}$ happens around $t=2261.88$ where $N(t+1)-N(t-1)=4$ . Consequently, on this interval

N(t+1)-N(t-1)<0.517869686\log t.

Moreover, for each $n$ the smallest value of $t$ for which $N(t+1)-N(t-1)=n$ is given in Table 4, from which one may bound $N(t+1)-N(t-1)$ with a step function, or any other function which happens to exceed that step function.

Table 4. Minimum value of

t

with

N(t+1)-N(t-1)=n

n	t
1	13.1347251417346937904572
2	48.7738324776723021819167
3	356.952685101632273755128
4	2261.87830538116111223015
5	27134.3628475733906424560
6	221227.766664702101313669
7	2603074.61468824424587333
8	21297085.9439615105210553
9	254721517.418748602610351
10	2786055796.5252751861828
11	29731208527.9429140229012
12	larger than 30610045999

(3.4)			$\displaystyle\left\|N_{\mathbb{Q}}(T)-\frac{T}{\pi}\log\left(\frac{T}{2\pi e}\right)\right\|$
			$\displaystyle\leq\left\|\frac{1}{4}-\frac{2}{\pi}\arctan\left(\frac{T}{\sigma_{1}-1}\right)-\frac{2}{\pi}\arctan(2T)\right\|+\|g(T)\|+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
			$\displaystyle\leq\left\|\frac{1}{4}-\frac{2}{\pi}\arctan\left(\frac{T}{\sigma_{1}-1}\right)-\frac{2}{\pi}\arctan(2T)\right\|+\frac{1}{25T}+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|$
			$\displaystyle=\|h(c,r,T)\|+\frac{1}{25T}+\frac{2}{\pi}\log\zeta\left(\sigma_{1}\right)+\frac{2}{\pi}\left\|\Delta_{\mathscr{C}_{2}}\arg((s-1)\zeta(s))\right\|,$

(3.19)		$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq\frac{1+\eta-\sigma}{\eta}\log\frac{c_{1}}{2^{c_{2}}}+\frac{\sigma-1}{\eta}\log\zeta(1+\eta)+\frac{1}{2}\log\left(\left(Q_{0}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
(3.19)			$\displaystyle+\frac{c_{2}(1+\eta-\sigma)}{\eta}\log\log\left(\left(Q_{0}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$
	$\displaystyle\leq\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)(\log 1.546)-\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)\log 2$
	$\displaystyle+\left(\frac{(2^{h+4}-1)(\sigma_{5+h}-\sigma)}{2(2^{h+4}-2)(\sigma_{5+h}-\sigma_{h+4})}+\frac{(2^{h+5}-1)(\sigma-\sigma_{h+4})}{2(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}\right)\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle+\left(\frac{(\sigma_{5+h}-\sigma)}{\sigma_{5+h}-\sigma_{h+4}}+\frac{(\sigma-\sigma_{h+4})}{\sigma_{5+h}-\sigma_{h+4}}\right)\log\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle=\log 1.546-\log 2+\log\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
	$\displaystyle+\left(\frac{(2^{h+4}-1)(\sigma_{5+h}-\sigma)}{2(2^{h+4}-2)(\sigma_{5+h}-\sigma_{h+4})}+\frac{(2^{h+5}-1)(\sigma-\sigma_{h+4})}{2(2^{h+5}-2)(\sigma_{5+h}-\sigma_{h+4})}\right)\log\left(\left(Q_{4+h,5+h}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq\frac{(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}(\log k_{1})+\frac{(\sigma-\frac{1}{2})}{\sigma_{4}-\frac{1}{2}}(\log 1.546)-\left(\frac{k_{3}(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}+\frac{(\sigma-\frac{1}{2})}{(\sigma_{4}-\frac{1}{2})}\right)\log 2$
		$\displaystyle+\left(\frac{(k_{2}+1)(\sigma_{4}-\sigma)}{2(\sigma_{4}-\frac{1}{2})}+\frac{15(\sigma-\frac{1}{2})}{28(\sigma_{4}-\frac{1}{2})}\right)\log\left(\left(Q_{2}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\left(\frac{k_{3}(\sigma_{4}-\sigma)}{\sigma_{4}-\frac{1}{2}}+\frac{(\sigma-\frac{1}{2})}{(\sigma_{4}-\frac{1}{2})}\right)\log\log\left(\left(Q_{2}+\sigma\right)^{2}+(\|t\|+T)^{2}\right).$

(3.20)	$\displaystyle\frac{1}{N}\log\left\|f_{N}(s)\right\|$	$\displaystyle\leq(1-2\sigma)\log\left(\frac{c_{1}}{2^{c_{2}+\frac{1}{2}}\sqrt{\pi}}\right)+2\sigma\log\frac{k_{1}}{2^{k_{3}}}+\frac{1}{2}\log\left((\sigma-1)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\frac{1-2\sigma+4k_{2}\sigma}{4}\log\left(\left(Q_{11}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$
		$\displaystyle+\left(c_{2}(1-2\sigma)+2k_{3}\sigma\right)\log\log\left(\left(Q_{11}+\sigma\right)^{2}+(\|t\|+T)^{2}\right)$

Improved estimates for the argument and zero-counting of Riemann zeta-function (With an appendix by Andrew Fiori)

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Theorem 1.2.

Remark.

Corollary 1.3.

Theorem 1.4.

Corollary 1.5.

Corollary 1.6.

Remark.

2. Preliminaries

2.1. Other useful results

Lemma 2.1 ([HSW21] Prop. 3.5).

Lemma 2.2 ([Leo24]).

Lemma 2.3.

Lemma 2.4 (Backlund’s trick).

Lemma 2.5 ([Ben+21]).

Lemma 2.6 (Phragmén-Lindelöf principle).

3. Proof of Theorem 1.2

Remark.

3.1. Convexity/subconvexity bounds and generalisation to nn-splitting of [12,1][\frac{1}{2},1]

Remark.

3.2. Estimating 1N​log⁡|fN​(s)|\frac{1}{N}\log|f_{N}(s)|

3.3. Defining Fc,r​(θ)F_{c,r}(\theta)

3.4. Conclusion

3.4.1. Constant C1C_{1}

3.4.2. Constants C2C_{2} and C2′C^{\prime}_{2}

3.4.3. Constants C3C_{3} and C3′C^{\prime}_{3}

4. Proof of Theorem 1.4

5. Proofs of Theorem 1.1 and Corollary 1.5

6. Proof of Corollary 1.6

Remark.

Acknowledgments

References

Appendix A A Numerical Study

A.1. Average Values

Observation A.1.

A.2. Range of Values

Proposition A.2.

Proof.

Observation A.3.

Theorem A.4.

Proof.

A.3. Extreme Values are Rare

Observation A.5.

Conjecture A.6.

A.4. Extreme Values are Common

Observation A.7.

A.5. Clusters of Zeros

Observation A.8.

Improved estimates for the argument and zero-counting of Riemann zeta-function
(With an appendix by Andrew Fiori)

3.1. Convexity/subconvexity bounds and generalisation to $n$ -splitting of $[\frac{1}{2},1]$

3.2. Estimating $\frac{1}{N}\log|f_{N}(s)|$

3.3. Defining $F_{c,r}(\theta)$

3.4.1. Constant $C_{1}$

3.4.2. Constants $C_{2}$ and $C^{\prime}_{2}$

3.4.3. Constants $C_{3}$ and $C^{\prime}_{3}$