The positivity technique and low-lying zeros of Dirichlet $L$-functions

We investigate several important quantities that naturally arise in the study of low-lying zeros of Dirichlet $L$-functions. Throughout this paper $\chi$ denotes a Dirichlet character with conductor $q>1$ unless otherwise specified. Let $|\gamma_{\chi}|$ (resp., $|\widetilde{\gamma_{\chi}}|$) be the height of the lowest (resp., lowest non-real) non-trivial zero of $L(s,\chi)$. That is,

Further, let $n_{\chi}=\underset{s=1/2}{\mathrm{ord}}L(s,\chi)$ be the order of vanishing of $L(s,\chi)$ at the central point. Siegel [Sie45, Theorem IV] (see also the last paragraph of [Sie45]) proved that $|\widetilde{\gamma_{\chi}}|\to 0$ as $q\to\infty$, and to be precise, that

Omar [Oma08] established the bound $n_{\chi}<\log q$ and verified Chowla’s conjecture, which asserts that $L(\frac{1}{2},\chi)\neq 0$ for all $\chi$ (originally stated only for real $\chi$ [Cho65]), for real primitive characters of modulus up to $10^{10}$. Conditionally on the generalized Riemann hypothesis (GRH), Omar [Oma08, Theorem 7] showed that

and used this to improve (1) to [Oma08, Theorem 11]

These results are achieved by the positivity technique, which involves applying a specialized form of Weil’s explicit formula to test functions with suitable properties, as will be explained in §2. This method was originally developed by Odlyzko and Serre, and subsequently improved by Poitou [Poi77], to furnish lower bounds on discriminants of number fields. We refer the reader to [Odl90] for a survey of its historical background and earlier development, and to [Mil02, Oma00, Oma14] for further applications to studying the first zeros of various families of $L$-functions.

For an alternative way to arrive at (2) and (3), one may start with the Riemann–von Mangoldt formula (see, e.g., [MV07, Corollary 14.6])

Here $N_{0}(t,\chi)$ counts the number of zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$ with $0<\beta<1$ and $-t\leq\gamma\leq t$ (any zero with ordinate exactly equal to $\pm t$ is counted with weight $1/2$), and

if $t\neq\gamma$ for any $\gamma$, otherwise $S(t,\chi):=\frac{1}{2}(S(t+0,\chi)+S(t-0,\chi))$. Under GRH, it is known that $S(t,\chi)\ll\frac{\log(q(|t|+1))}{\log\log(q(|t|+3))}$ uniformly in $q$ and $t$ due to a result of Selberg [Sel46]. Taking $t\to 0$ on the right-hand side of (4) gives (2), and so $N(t,\chi)>n_{\chi}$ when $t\gg(\log\log q)^{-1}$, proving (3). Using the theory of Beurling–Selberg type extremal functions, Carneiro, Chandee and Milinovich [CCM15, Theorem 6 & Example 8] sharpened the conditional bound on $S(t,\chi)$ to

Consequently,

and

By the same reasoning as above, (6) implies that

Much more work has been done towards understanding the distribution of low-lying zeros in families of Dirichlet $L$-functions, for example the family of quadratic Dirichlet $L$-functions and the family of all Dirichlet $L$-functions of a fixed modulus. In the latter direction, Murty [Mur90] showed under GRH that

essentially by using the positivity technique. As a result, for any $\epsilon>0$ the proportion of $\chi$ mod $q$ such that $L(\frac{1}{2},\chi)\neq 0$ is at least $\frac{1}{2}-\epsilon$ for all sufficiently large $q$. By constructing certain extremal functions in reproducing kernel Hilbert spaces, Carneiro, Chirre and Milinovich [CCM22] improved on the average order of vanishing at low-lying heights of the form $\frac{2\pi t}{\log q}$ for $t>0$, but at $t=0$ (9) has not been updated so far ¹¹1For the family of primitive Dirichlet $L$-functions with conductors $\in[Q,2Q]$, the non-vanishing proportion at the central point has been extended beyond $\frac{1}{2}$ for large $Q$, unconditionally by Pratt [Pra19] and conditionally by Drappeau et al. [DPR23].. The best unconditional result in this direction is due to Bui [Bui12] who proved that $L(\frac{1}{2},\chi)\neq 0$ for at least $34\%$ of the primitive characters modulo $q$. Restricting to prime moduli, Khan, Milićevié and Ngo [KMN22] improved this proportion to $\frac{5}{13}-o(1)$.

In their seminal work [HR03], Hughes and Rudnick established, assuming GRH, the one-level density for the unitary family of Dirichlet $L$-functions modulo $q$ for test functions $f$ with $\mathrm{supp}\>(\widehat{f})\in[-2,2]$. As a corollary, they [HR03, Theorem 8.1] showed that

(The restriction of $q$ to primes is to ensure that each non-principal $\chi$ mod $q$ is primitive, which simplifies the discussion.) In other words, there exists zeros as low as $\frac{1}{4}+o(1)$ of the average spacing between consecutive zeros. Using the variance of the one-level density, they [HR03, Theorem 8.3] also proved that for any $\beta\geq 0.633$,

For instance, we see by taking $\beta=1$ that at least $80\%$ of the characters modulo $q$ have low-lying zeros within the expected height, and this proportion increases to $\frac{11\pi^{2}-3}{12\pi^{2}}=0.8913\cdots$ as $\beta\to\infty$.

The purpose of this work is further develop Omar’s argument in [Oma08] to reprove and refine some of the aforementioned results, in hopes of demonstrating the versatility and efficacy of the positivity technique, which is at the same time relatively simple and straightforward. For individual Dirichlet $L$-functions, we refine (6), (7) and (8) by removing a $\log\log\log q$ factor from each error term. In particular, this argument circumvents the intricate analysis of $S(t,\chi)$ and thus has the additional advantage that it can easily be made fully explicit (see Theorem 16).

This result indicates that if (12) turns out to be optimal for some primitive $\chi\bmod q$ (although this is not the case in all likelihood), then for all $C>\log 4+1$, the average spacing between zeros with height

is at most $(\frac{8}{\pi}+o(1))\frac{1}{\log q}<\frac{2\pi}{\log q}$.

In fact, (5)-(8) are special cases of results in [CCM15] that are applicable to a larger class of $L$-functions including those associated to cuspidal automorphic representations of $\mathrm{GL}_{m}$ over a number field. These results sharpen what Omar obtained in this general setting [Oma14]. Thus, for completeness, we shall give extensions of Theorems 1–3 in §7.

Slightly more precise versions of (9) and (10) are also obtained for the family of $\chi$ mod $q$.

Notice that the restriction of $q$ to primes is removed from (10). A similar estimate holds for $|\widetilde{\gamma_{\chi}}|$ with $1/4$ replaced by $1/2$ on the right-hand side.

In the other direction, we prove the following result on large heights of the first zeros:

Regarding the proportion of characters with low-lying zeros, we improve (11) for all $1/2<\beta<\infty$.

Without any constraints on the modulus $q$, we give in Remark 4 a slightly smaller proportion that is $1-O(\beta^{-1})$ as $\beta\to\infty$. These results imply that although we are unable to rule out the existence of $\chi\bmod q$ satisfying $|\gamma_{\chi}|\gg(\log\log q)^{-1}$ (see Theorem 1), the proportion of such characters is $o(1)$ as $q\to\infty$. A similar estimate for the family of quadratic Dirichlet $L$-functions is obtained in Theorem 14.

For the proof, in contrast to Hughes and Rudnick’s approach (of fixing some test function and applying Chebyshev’s inequality using the variance of the one-level density), we pursue a more direct cardinality argument based on the Cauchy–Schwarz inequality and allow for flexibility with the choice of the test function (see §4.6).

We begin by stating the version of Weil’s explicit formula as formulated by Barner [Bar81] and adapted for $L(s,\chi)$. We say that a function $F:\mathbb{R}\to\mathbb{R}$ is admissible if it is even and satisfies $F(0)=1$ along with the following properties:

1. (a)

   $F$ is continuously differentiable everywhere except at finitely many points $a_{i}$ for which $F(a_{i})=\frac{1}{2}(F(a_{i}+0)+F(a_{i}-0))$.

2. (b)

   There exists $\epsilon>0$ such that $F(x)$ and $F^{\prime}(x)$ are both $O(e^{-(1/2+\epsilon)|x|})$ as $x\to\infty$.

Put

Then we have the following:

As usual, $\Lambda(n)$ denotes the von Mangoldt function, which takes on the value $\log p$ when $n=p^{m}$ is a prime power, and 0 otherwise.

Define the Fourier transform of an admissible function $F$ by

Note that if $\rho=\frac{1}{2}+i\gamma$, then $\Phi(F)(\rho)=\widehat{F}(\gamma)$ and $\Phi(F_{T})(\rho)=T\widehat{F}(T\gamma)$ where $F_{T}(x):=F(x/T)$. We assume GRH henceforth, so that every non-trivial zero is of this form.

To see how to bound $|\gamma_{\chi}|$ from above, suppose that $\widehat{F}(t)\geq 0$ for $|t|\leq T_{0}$ and that $\widehat{F}(t)\leq 0$ for $|t|\geq T_{0}$. Putting $T=T_{0}/|\gamma_{\chi}|$ (we may assume that $|\gamma_{\chi}|\neq 0$), we have $\Phi(F_{T})(\rho)\leq 0$ for each $\rho$. If we can estimate the second and third terms on the right-hand side of (14) in terms of $T$, then for sufficiently large $q$ this will lead to a lower bound on $T$ in terms of $q$, and hence an upper bound on $|\gamma_{\chi}|$.

The treatment for $|\widetilde{\gamma_{\chi}}|$ is similar, but we get a weaker bound because the zeros at $\frac{1}{2}$ do make a positive contribution to the sum. To make this precise, we first derive an upper bound on $n_{\chi}$ by applying (14) to $H_{T}$ for some $H$ with non-negative Fourier transform $\widehat{H}$. By dropping the contribution from all zeros $\rho\neq\frac{1}{2}$, we see that the left-hand side is at least $n_{\chi}\widehat{H}(0)T$, and the problem therefore reduces to bounding the right-hand side and choosing the optimal $T$.

Finally, when considering the family of all non-principal $\chi$ mod $q$ simultaneously, we sum both sides of (14) over $\chi$ and exploit the orthogonality of characters instead of trivially bounding the sum over prime powers by the triangle inequality.

We now describe two classes of admissible functions that could be used to derive an upper bound on $|\gamma_{\chi}|$, for which we require that $\widehat{F}(t)\leq 0$ outside of some bounded interval. For practical reasons we also need $F$ to be compactly supported. The first has the form

where $\alpha>1$. (Omar fixes $\alpha=3$ in [Oma08], while our $\alpha$ will be a parameter depending on $q$.) One can check that

so that $\widehat{F^{\alpha}}(t)\leq 0$ for $|t|\geq\sqrt{\frac{\alpha+1}{\alpha-1}}\pi$.

Another example is given by the Fourier transforms of the Beurling–Selberg minorants for symmetric intervals $[-\beta,\beta]$ with $\beta>1/2$. In the 1930s, Beurling studied the entire functions

and observed that $B^{-}(x)\leq\mathrm{sgn}(x)\leq B^{+}(x)$ for all $x\in\mathbb{R}$, that $\int_{-\infty}^{\infty}|B^{\pm}(x)-\mathrm{sgn}(x)|\,\mathrm{d}{x}=1$, and that $\widehat{B^{\pm}}(t)$ ²²2Here we interpret $\widehat{B^{\pm}}(t)$ in the sense of distributions as $B^{\pm}\not\in L^{1}(\mathbb{R})$. has compact support in $[-2\pi,2\pi]$. The Beurling–Selberg minorant for $[-\beta,\beta]$ is then given by

One can show that $\widehat{B^{-}_{[-\beta,\beta]}}(0)=2\beta-1$ (see §4.5). Now define for $\beta>1/2$

It is not hard to verify that $J^{\beta}(0)=1$, that $J^{\beta}(x)$ is even and compactly supported in $[-2\pi,2\pi]$, and that $\widehat{J^{\beta}}(t)\leq\mathbbm{1}_{[-\beta,\beta]}(t)$, which denotes the characteristic function on $[-\beta,\beta]$. (In particular, $\widehat{J^{\beta}}(t)\leq 0$ for all $|t|\geq\beta$.) See [Vaa85] for a detailed exposition on the constructions and some earlier applications of such functions.

To end this section we introduce two lemmas concerning the term $I_{\chi}$ and the sum over prime powers that appear in the explicit formula (14) when applied to the test function $F^{\alpha}$ as defined in (16). The proofs can easily be adapted for other test functions.

Assume that $L(\frac{1}{2},\chi)\neq 0$, i.e., $|\gamma_{\chi}|\neq 0$, since the theorem trivially holds otherwise. Consider the function $F^{\alpha}_{T}(x)=F^{\alpha}(x/T)$ with $F^{\alpha}$ from (16) and $T=\sqrt{\frac{\alpha+1}{\alpha-1}}\pi/|\gamma_{\chi}|$. We see from (3) that $T\gg\log\log q$. Note that $\sum_{\rho}\Phi(F^{\alpha}_{T})(\rho)=\sum_{\gamma}T\widehat{F^{\alpha}}(T\gamma)\leq 0$ because $\widehat{F^{\alpha}}(t)\leq 0$ for all $|t|\geq\sqrt{\frac{\alpha+1}{\alpha-1}}\pi$. Thus, in view of Theorem 8 and Lemmas 9 and  10, we have

Set $\alpha=\log\log q$ and

so that $\frac{e^{T/2}}{T/2}\geq\Delta$. Then

On expressing $T$ in terms of $|\gamma_{\chi}|$ and $\alpha$ and appealing to the fact that $\sqrt{1-x}=1-x/2+O(x^{2})$ as $x\to 0$, we arrive at

which completes the proof of (12).

Next we prove the second assertion regarding the number of zeros below $t_{1}:=\frac{\pi}{2}(\log\log q)^{-1}+\frac{\pi}{2}C(\log\log q)^{-2}$ with $C>\log 4+1$. Let $n$ be the positive integer such that $|\gamma_{\chi,n}|<t_{1}\leq|\gamma_{\chi,n+1}|$ (here $\gamma_{\chi,k}$ denotes the $k^{\text{th}}$ zero ordered by height), and our goal is to show that

Take $\alpha=a\log\log q$ for some parameter $a>0$. For $T:=\sqrt{\frac{\alpha+1}{\alpha-1}}\pi/t_{1}=2(\log\log q-C+1/a+o(1))$, we have $\Phi(F^{\alpha}_{T})(\rho_{k})\leq 0$ for all $k>n$, and then another application of (14) gives

After rearranging the inequality we find that

Finally, it is a calculus exercise to determine that the optimal choice of $a$ is $(C-\log 4)^{-1}$, which yields (22).

As in [Oma08, Lemma 9], we work with the admissible function