Explicit bounds for the Riemann zeta function
and a new zero free region

Let $\zeta(s)$ be the Riemann zeta function, with $s=\sigma+it$ a complex variable. It is known that all the non trivial zeros of $\zeta(s)$ have real part $\sigma\in(0,1)$. Detecting zero free regions for $\zeta(s)$ inside the critical strip $0<\sigma<1$ is an open problem that has always caught much interest in analytic number theory. Great effort has been put in trying to find both asymptotically and explicit regions inside the critical strip where there are no zeros of $\zeta(s)$. The classical zero-free region is of the form $\sigma>1-1/(R_{0}\log|t|)$, where $R_{0}$ is a positive constant. The best known result of this form is due to Mossinghoff, Trudgian and Yang [MTY22] with $R_{0}=5.558691$ for every $|t|\geq 2$ (see [Ste70, RS75, Kon77, Kad05, JK14, MT14] for previous results). Littlewood zero-free region [Lit22] is instead of the form of $\sigma>1-\log\log|t|/C_{1}\log|t|$, where $C_{1}$ is a positive constant. It has been made first explicit by Yang [Yan23], who found $C_{1}=21.432$ for $|t|\geq 3$.
Asymptotically larger zero-free regions for the Riemann zeta function $\zeta(s)$, known as Korobov–Vinogradov zero-free regions, are of the form

where $C_{2}$ is a positive constant and are due to the method of Korobov [Kor58] and Vinogradov [Vin58], in which the main tool is an upper bound on $|\zeta(\sigma+it)|$ when $\sigma$ is near to the line $\sigma=1$. Upper bounds of this form were first made explicit by Richert [Ric67], who used Korobov-Vinogradov method to prove that

with $B=100$ and $A$ a certain absolute constant. Although smaller values for $B$ were already found (see [Kul99]), the first completely explicit bound of the form (1.2) is due to Cheng [Che99], with $A=175$ and $B=46$. This estimate was further improved in 2002 by Ford [For02a] to $A=76.2$ and $B=4.45$ for every $|t|\geq 3$ and $1/2\leq\sigma\leq 1$.
Explicit bounds of the form (1.2) play a fundamental role in detecting explicit Korobov–Vinogradov zero-free regions for $\zeta(s)$. There are several results regarding the value of the constant $C_{2}$ in (1.1) and the current best known estimate is due to Mossinghoff, Trudgian and Yang [MTY22], with $C_{2}=55.241$ for every $|t|\geq 3$ (see [Che00, For02, For22] for previous results).
In this paper, we will improve the values for both the constants $A,B$ in (1.2) and, as a consequence, we find an improved Korobov-Vinogradov zero-free region for $\zeta(s)$. More precisely, denoting with $\zeta(s,u)$ the Hurwitz zeta function $\zeta(s,u)=\sum_{n=0}^{\infty}(n+u)^{-s}$ defined for every $\Re s>1$ and $0<u\leq 1$, we will prove the following result for both $\zeta(s)=\zeta(s,1)$ and a generic Hurwitz function $\zeta(s,u)$.

The bound on $\zeta(s,u)$ found in Theorem 1.1 might be useful to bound Dirichlet $L$-functions due to the relation $L(s,\chi)=q^{-s}\sum_{m=1}^{q}\chi(m)\zeta(s,m/q)$, where $\chi$ is a Dirichlet character modulo $q$.
Although the new values $A=70.6995$ and $B=4.43795$ found in Theorem 1.1 are modest improvements on those found by Ford in [For02a] ($A=76.2$ and $B=4.45$ respectively), their importance relies on the fact that just a small improvement for both $A$ and $B$ can lead to improvements in several other results in analytic number theory. In particular, they have many applications in finding improved estimates for Korobov–Vinogrado zero-free region and in estimating the error term in the prime number theorem, both in an effective and ineffective way.
Furthermore, improvements on $A$ and $B$ only of the size found in Theorem 1.1 were already expected. In [For02a] (Section 8, point $1$), Ford already predicted that an optimization of the argument involving the Vinogradov integral would have lower $B$ by less than $0.02$, which is consistent with our new value for $B=4.43795$ found in Theorem 1.1.

As already mentioned, an immediate consequence of Theorem 1.1 is a new explicit zero free region for the Riemann zeta function.

Theorem 1.1 has also an influence on asymptotically Korobov-Vinogradov zero-free regions of the form

where $c$ is a positive constant for $|t|$ sufficiently large. More precisely, we get $c=48.0718$, improving $c$ on the current best value of $48.1588$ due to Mossinghoff, Trudgian and Yang [MTY22] (see [For02] for previous results).

As in [MTY22], the proofs of Theorem 1.2 and Theorem 1.3 rely on some non-negative trigonometric polynomials. We recall that, for every $K\geq 2$, a $K$-th degree non-negative trigonometric polynomial $P_{K}(x)$ is defined as

where $b_{k}$ are constants such that $b_{k}\geq 0,b_{1}>b_{0}$ and $P_{K}(x)\geq 0$ for all real $x$.
Finally, one can use Theorem 1.3 to improve on the error term in the prime number theorem. As per Ford [For02a], we get the following estimate for the error term in the prime number theorem

with

where $c$ is the constant in (1.4). Using the new value for $c$ found in Theorem 1.3 we obtain $d=0.212579$, which is a slightly improvement on $d=0.2123$ found by Mossinghoff, Trudgian and Yang [MTY22].

As in [For02a], Theorem 1.1 follows from a uniform upper bound on the sum

where $N$ is a positive integer and $t\geq N$. This upper bound is of the form

with $u,c>0$ and $\lambda=\log t/\log N.$ The strength of Ford’s argument in [For02a] relies on some explicit estimates for both the Vinogradov integral and a quantity that counts the number of solutions of incomplete Diophantine systems that, combined with estimates for the exponential sum $S(N,t)$, give a completely explicit uniform upper bound on $S(N,t)$. Explicit bounds for the Vinogradov integral were further improved by Preobrazhenskiĭ [Pre11] in 2011 and Steiner in 2019 [Ste19] (see [Hua49, Ste70, Tyr87, ACK04] for previous results).
We recall that the Vinogradov integral is defined as

where $\boldsymbol{\alpha}=\left(\alpha_{1},\ldots,\alpha_{k}\right)$ and $e(z)=e^{2\pi iz}$, or equivalently, $J_{s,k}(P)$ is defined as the number of solutions of the simultaneous equations

Regarding incomplete systems, we denote with $J_{s,k,h}(\mathscr{B})$ the number of solutions of the system

where $\mathscr{B}$ is a suitable set. For our purpose, we will use the definition given by Ford in [For02a] $\mathscr{B}=\mathscr{C}(P,R)$ where $\mathscr{C}(P,R)$ is the set of integers $\leq P$ composed only of prime factors in $(\sqrt{R},R]$. Hence, $J_{s,k,h}(\mathscr{C}(P,R))$ is defined as

where

and $\boldsymbol{\alpha}=\left(\alpha_{h},\cdots,\alpha_{k}\right)$.
In his paper [For02a], Ford estimated $S(N,t)$ working separately and with different techniques on three different ranges of $\lambda$. More precisely, he considered the ranges $\lambda\leq 87$, $87\leq\lambda\leq 220$ and $\lambda\geq 220$, where the critical case happens to be around $\lambda=87$. However, it is possible to shift the critical case to $\lambda=84$, with a consequent improved upper bound on $S(N,t)$ and so improved values of $A$ and $B$. Indeed, in his paper [For02a] (p. 590), Ford stated a variant of his argument to find even better estimates for the Vinogradov integral when $k$ is small. These results, combined with Preobrazhenskiĭ’s argument for $k\geq 90000$ in [Pre11] and some explicit bounds for the Vinogradov integral when $s\ll k^{2}$ due to Tyrina [Tyr87], will give the following explicit bounds for $J_{s,k}$.

Using these new bounds for the Vinogradov integral we will prove the following result.

In proving both Theorem 1.4 and Theorem 1.5, we tried to make near-optimal choices for all parameters that are used in the argument. Hence, it seems unlikely that a substantial improvement in the $B$ constant in Theorem 1.1 can be obtained via better choices of parameters alone.

An immediate corollary of Theorem 1.5 involving Dirichlet characters is the following one.

The proof is the same as that of Corollary 2A in [For02a].

We recall some preliminary results we will use later in the proof of the theorems.
First of all, given a fixed $k$, suppose $0\leq d\leq k-1$ and $T$ is a positive integer. The $k$-tuple of polynomials $\boldsymbol{\Psi}=\left(\Psi_{1},\ldots,\Psi_{k}\right)\in\mathbb{Z}[x]^{k}$ is said to be of type $(d,T)$ if $\Psi_{j}$ is identically zero for $j\leq d$, and for some integer $m\geq 0$, when $j>d,\ \Psi_{j}$ has degree $j-d$ with leading coefficient $\frac{j!}{(j-d)!}2^{m}T$.
Then, let $K_{s,d}(P,Q;\boldsymbol{\Psi};q)$ be the number of solutions of

with $1\leq z_{i},w_{i}\leq P$ and $1\leq x_{i},y_{i}\leq Q$. Also, let $L_{s,d}(P,Q;\boldsymbol{\Psi};p,q,r)$ be the number of solutions of

with $1\leq z_{i},w_{i}\leq P$, $z_{i}\equiv w_{i}\left(\bmod\ p^{r}\right)$ and $1\leq u_{i},v_{i}\leq Q$.

As per Ford [For02a], Lemma 2.1 and Lemma 2.2 imply the following result.

We will use the definition of $\phi_{J}$ given in Lemma 2.3 for the range $k<90000$. However, when $k$ becomes large, the influence that the new funcitons $\Phi_{j}$ have on estimating the Vinogradov’s integral becomes negligible, as already pointed out by Ford in [For02a] (p. 590). Hence, for $k\geq 90000$, we use the following result used by Ford in his work.

For a given $k,r$ and $\Delta$, we let $\delta_{0}(k,r,\Delta)$ be the value of $\Delta^{\prime}$ coming from Lemma 2.3 or Lemma 2.4, where we take $j$ maximal satisfying (2.1).

As already mentioned before, some explicit bounds for the Vinogradov’s integral were found by Tyrina [Tyr87], and they give evident improvements when $s\ll k^{2}$. In her argument, Tyrina defines recursively sequences $r_{n},s_{n}$, and $\Delta_{s_{n}}$, with $s_{1}=k$ and $\Delta_{s_{1}}=k$. More precisely, for $s\geq 1$ we take $r_{n}$ to be the integer nearest to the number

and we define recursively

where

Then, we consider the function $x(y),\ k\leq y<k(k+1)/2$ :

This function increases monotonically and assumes the values from $k$ to infinity as $y$ increases from $k$ to $k(k+1)/2$. The function $y=y(x)$ which is inverse to $x(y)$ is also monotonically increasing. Finally, let $l_{0}$ denote the integer $\lceil x\left(k^{2}/2\right)\rceil$, where

Tyrina proved the following result.

New explicit bounds for $J_{s,k}$ were found by Ford in 2002 [For02a].

In 2011, Preobrazhenskiĭ [Pre11] improved the value of $\rho$ in Theorem 2.7 when $k\geq 16000$ to $\rho=3.213303$. Some further explicit bounds for the Vinogradov’s integral were found also by Steiner [Ste19] using Wooley’s efficient congruencing method [Woo12, Woo13, Woo17]. He proved that for every $k\geq 3$, $s\geq\frac{5}{2}k^{2}+k$, and $P\geq s^{10}$, we have $J_{s,k}(P)\leq CP^{2s-\frac{1}{2}k(k+1)}$, where $C$ is roughly $k^{k^{O(\log(k)/\log(\lambda))}}$. However, although the exponent of $P$ is the optimal one, the constant $C$ is far too large compared to that one found by Ford in Theorem 2.7, which is of order $k^{O(k^{3})}$. In order to have a reasonable estimate for $J_{s,k}$, one should find a constant that is at most of order $k^{O(k^{4})}$, since otherwise, after having taken the $k^{4}-$th root (this passage is required by the argument in Section 4, where the constant appearing in the upper bound for the Vinogradov integral will be elevated to the power of $1/(2rs)$, with both $r$ and $s$ of order $O(k^{2})$), the constant cannot be controlled. Hence, as already Steiner pointed out in his paper ([Ste19], p. 359), the estimate he found for $J_{s,k}$ cannot be applied to Ford’s argument.

In [For02a], Ford found also an upper bound for the incomplete systems defined by (1.9).

Finally, we recall a few results found in [For02a] involving both $S(N,t)$ defined in (1.7) and some bounds for both $\zeta(s)$ and $\zeta(s,u)$ that we will use to prove Theorem 1.5 and Theorem 1.1 respectively.