1·1. Main results

Let $\zeta(s)$ be the Riemann zeta function, and let $N(\sigma,T)$ denote the number of zeros $\beta+i\gamma$ of $\zeta(s)$ with $\beta\geq\sigma\geq 0$ and $|\gamma|\leq T$ . The asymptotic

(1·1) \begin{equation} N(T)\;:\!=\; N(0,T)\sim (1/\pi)T\log T\end{equation}

follows from the argument principle. The Riemann hypothesis (RH) asserts that $\zeta(s)\neq 0$ for $\textrm{Re}(s)>{1}/{2}$ , so $N(\sigma,T)=0$ for $\sigma>{1}/{2}$ . Selberg [ Reference TitchmarshTit86 , theorem 9·19C] proved a delicate zero density estimate that recovers the upper bound in (1·1) at $\sigma={1}/{2}$ , namely

(1·2) \begin{equation} N(\sigma,T)\ll T^{1-\frac{1}{4}(\sigma-\frac{1}{2})}\log T,\qquad \sigma\geq \tfrac{1}{2}.\end{equation}

(See Baluyot [ Reference Alan and BaluyotBal17 , theorem 1·2·1] for an improvement.) Selberg’s estimate implies that

\begin{equation*}\frac{1}{N(T)}\#\Big\{\rho=\beta+i\gamma\colon \zeta(\rho)=0,\;|\gamma|\leq T,\;\beta\in\Big[\frac{1}{2}-\frac{4\log\log T}{\log T},\frac{1}{2} +\frac{4\log\log T}{\log T}\Big]\Big\}\ll \frac{1}{\log T}.\end{equation*}

As an application of (1·2), Selberg proved a central limit theorem for $\log|\zeta({1}/{2}+it)|$ :

(1·3) \begin{equation}\lim_{T\to\infty}\frac{1}{T}{\text{meas}}\Big\{T\leq t\leq 2T\colon \log|\zeta(\tfrac{1}{2}+it)| \geq V\sqrt{\tfrac{1}{2}\log\log T}\Big\}\sim\frac{1}{\sqrt{2\pi}}\int_V^{\infty}e^{-u^2/2}du.\end{equation}

Let $\mathscr{A}$ be the set of cuspidal automorphic representations of $\textrm{GL}_2$ over $\mathbb{Q}$ with unitary central character. For $\pi\in\mathscr{A}$ , let $L(s,\pi)$ be its standard L-function. Define

\begin{equation*} N_{\pi}(\sigma,T)\;:\!=\;\#\{\rho=\beta+i\gamma\colon \beta\geq\sigma,\;|\gamma|\leq T,\;L(\rho,\pi)=0\}.\end{equation*}

As with $\zeta(s)$ , the argument principle can be used to prove that

(1·4) \begin{equation} N_{\pi}(T)\;:\!=\; N_{\pi}(0,T)\sim (2/\pi)T\log T.\end{equation}

The generalised Riemann hypothesis (GRH) asserts that $L(s, \pi) \neq 0$ for $\textrm{Re}(s) > {1}/{2}$ . Jutila [ Reference JutilaJut83 ] and Selberg [ Reference SelbergSel92 ] observed that analogues of (1·2) and (1·3) should also hold for Hecke–Maaß newforms. When $\pi \in \mathscr{A}$ corresponds with a holomorphic cuspidal newform of even weight $k \geq 2$ , Luo [ Reference LuoLuo95 , theorem 1·1] and Li [ Reference Ford and ZaharescuFZ15 , section 7] proved that $N_{\pi}(\sigma, T) \ll T^{1-\frac{1}{72}(\sigma-\frac{1}{2})}\log T$ . We prove:

Corollary 1·2. Let $\pi \in \mathscr{A}$ . If $V\in\mathbb{R}$ , then as $T\to\infty$ , we have:

\begin{equation*} \lim_{T\to\infty}\frac{1}{T}{\text{meas}}\Big\{T\leq t\leq 2T\colon \log|L(\tfrac{1}{2}+it,\pi)| \geq V\sqrt{\tfrac{1}{2}\log\log T}\Big\}=\frac{1}{\sqrt{2\pi}}\int_V^{\infty}e^{-u^2/2}du; \end{equation*}

\begin{equation*} \lim_{T\to\infty}\frac{1}{T}{\text{meas}}\Big\{T\leq t\leq 2T\colon \arg L(\tfrac{1}{2}+it,\pi) \geq V\sqrt{\tfrac{1}{2}\log\log T}\Big\}=\frac{1}{\sqrt{2\pi}}\int_V^{\infty}e^{-u^2/2}du. \end{equation*}

We study the distribution of imaginary parts of the nontrivial zeros of $L(s, \pi)$ using Theorem 1·1. First, we give an incomplete history of such results for $\zeta(s)$ . Hlawka [ Reference HlawkaHla75 ] proved that if $\alpha\in\mathbb{R}$ is fixed and $h\colon \mathbb{T}\to\mathbb{C}$ is continuous, then

(1·5) \begin{equation} \lim_{T\to\infty}\frac{1}{N(T)}\sum_{|\gamma|\leq T}h(\alpha\gamma)=\int_{\mathbb{T}}h(t)dt,\end{equation}

where $\gamma$ varies over the imaginary parts of the nontrivial zeros of $\zeta(s)$ and $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ . Thus the sequence of fractional parts $\{\gamma\alpha\}$ is equidistributed modulo 1. However, given a rate of convergence, there exist continuous functions h such that the limit in (1·5) cannot be attained with said rate (see also [ Reference Ford and ZaharescuFZ05 , theorem 7]). Therefore, (1·5) is the best that one can say for arbitrary h.

Ford and Zaharescu [ Reference Ford and ZaharescuFZ05 , corollary 2] established the existence of a second order term, proving that if $h\;:\;\mathbb{T}\to\mathbb{C}$ is twice continuously differentiable,Footnote ¹ then

(1·6) \begin{equation} \sum_{|\gamma|\leq T}h(\alpha\gamma)=N(T)\int_{\mathbb{T}}h(t)dt+T\int_{\mathbb{T}}h(t)g_{\alpha}(t)dt+o(T),\end{equation}

where

(1·7)

Despite the limitations on the analytic properties of h, Ford and Zaharescu still conjectured [ Reference Ford and ZaharescuFZ05 , conjecture A] that for any interval $\mathbb{I}\subseteq\mathbb{T}$ of length $|\mathbb{I}|$ , we have

(1·8) \begin{equation}\sum_{\substack{|\gamma|\leq T \\ \{\alpha\gamma\}\in\mathbb{I}}}1 = |\mathbb{I}|N(T)+T\int_{\mathbb{I}}g_{\alpha}(t)dt+o(T),\end{equation}

which implies that

(1·9) \begin{equation}D_{\alpha}(T)\;:\!=\;\sup_{\mathbb{I}\subseteq \mathbb{T}}\Big|\frac{1}{N(T)}\sum_{\substack{|\gamma|\leq T \\ \{\alpha\gamma\}\in\mathbb{I}}}1 -|\mathbb{I}|\Big|=\frac{T}{N(T)}\sup_{\mathbb{I}\subseteq\mathbb{T}}\Big|\int_{\mathbb{I}}g_{\alpha}(t)dt\Big|+o\Big(\frac{1}{\log T}\Big).\end{equation}

Ford, Soundararajan and Zaharescu [ Reference Ford, Soundararajan and ZaharescuFSZ09 ] made some progress toward the conjectured asymptotics (1·8) and (1·9). Unconditionally, they proved that

(1·10) \begin{equation}D_{\alpha}(T)\geq \frac{T}{N(T)}\sup_{\mathbb{I}\subseteq\mathbb{T}}\Big|\int_{\mathbb{I}}g_{\alpha}(t)dt\Big|+o\Big(\frac{1}{\log T}\Big).\end{equation}

Assuming RH, they proved that

(1·11) \begin{equation}\Big|\sum_{\substack{|\gamma|\leq T \\ \{\alpha\gamma\}\in\mathbb{I}}}1 - |\mathbb{I}|N(T)-T\int_{\mathbb{I}}g_{\alpha}(t)dt\Big|\leq \Big(\frac{\alpha}{2}+o(1)\Big)T.\end{equation}

Along with making some appealing connections between the conjectured asymptotics (1·8) and (1·9) and other intriguing open problems like pair correlation of zeros of $\zeta(s)$ and the distribution of primes in short intervals, they proved analogues for other L-functions of (1·10) (assuming a zero density estimate of the form (1·2)) and (1·11) (assuming GRH).

In this paper, we extend the work in [ Reference Ford and ZaharescuFZ05, Reference Ford, Soundararajan and ZaharescuFSZ09, Reference Ford, Meng and ZaharescuFMZ17, Reference Liu and ZaharescuLZ21 ] to $L(s,\pi)$ for any $\pi\in\mathscr{A}$ using Theorem 1·1. Let $n \ge 1$ . Consider the $\boldsymbol{\alpha}\in\mathbb{R}^n$ for which there exists a constant $C_{\boldsymbol{\alpha}}>0$ such thatFootnote ²

(1·12) \begin{equation} |\boldsymbol{m} \cdot \boldsymbol{\alpha}| \geq C_{\boldsymbol{\alpha}}e^{-\|\boldsymbol{m}\|_2 }\qquad {\text{for all}} \, {\boldsymbol{m}\in\mathbb{Z}^n \setminus \{\textbf{0}\}$},\end{equation}

where $\|\boldsymbol{m}\|_p$ is the $\ell^p$ norm on $\mathbb{R}^n$ for $1\leq p\leq\infty$ . This is a technical artifact of our extension to $\mathbb{R}^n$ ; when $n=1$ , the condition reduces to $\alpha\neq 0$ . Our density function $g_{\pi, \boldsymbol{\alpha}}(\boldsymbol{t})$ , which extends (1·7) for $n\geq 2$ , is identically zero unless there exists a matrix $M=(b_{jk}) \in \mathcal{M}_{r\times n}(\mathbb{Z})$ with linearly independent row vectors $\boldsymbol{b}_j$ and $\gcd(b_{j1}, \ldots, b_{jn}) = 1$ for all $1 \leq j \leq r$ ; fully reduced rationals $a_1/q_1, \ldots, a_r/q_r$ ; and distinct primes $p_1, \ldots, p_r$ such that

(1·13) \begin{equation} M\boldsymbol{\alpha}^{\intercal}=\left(\frac{a_1}{q_1}\frac{\log p_1}{2\pi},\ldots,\frac{a_r}{q_r}\frac{\log p_r}{2\pi}\right)^{\intercal}.\end{equation}

Among such possible matrices M, choose one with maximal r, which uniquely determines the row vectors $\boldsymbol{b}_j=(b_{j1},\ldots,b_{jn})$ . If such an M exists, then define

(1·14) \begin{equation} g_{\pi, \boldsymbol{\alpha}}(\boldsymbol{t}) \;:\!=\; -\frac{2}{\pi} \Re\sum_{j = 1}^{r} \sum_{l = 1}^{\infty} \frac{\Lambda_{\pi}({p_j}^{a_jl})}{{p_j}^{a_jl/2}} e^{-2 \pi i q_j l ({\boldsymbol{b}_\boldsymbol{j}} \cdot \boldsymbol{t})},\end{equation}

where

\begin{equation*} -\frac{L^\prime(s,\pi)}{L(s,\pi)} = \sum_{n =1}^{\infty} \frac{\Lambda_{\pi}(n)}{n^s},\qquad \textrm{Re}(s)>1.\end{equation*}

Let $\pi\in\mathscr{A}$ , $n\geq 1$ , and $\mathbb{B} \subseteq \mathbb{T}^n$ be a product of n subintervals of $\mathbb{T}$ . The conjecture in (1·8) can be extended to $\pi\in\mathscr{A}$ as follows:

(1·15) \begin{equation}\sum_{\substack{|\gamma|\leq T \\ \{\gamma \boldsymbol{\alpha}\} \in \mathbb{B}}} 1 = \textrm{vol}(\mathbb{B}) N_{\pi}(T) + T \int_{\mathbb{B}}g_{\pi, \boldsymbol{\alpha}}(\boldsymbol{t}) {d}\boldsymbol{t} + o(T),\end{equation}

where $\gamma$ ranges over the imaginary parts of the nontrivial zeros of $L(s,\pi)$ . As progress toward (1·15), we prove an unconditional n-dimensional version of (1·6) for $\textrm{GL}_2$ L-functions. In what follows, let $C^u(\mathbb{T}^n)$ be the set of u-times continuously differentiable functions $h\colon\mathbb{T}^n\to\mathbb{R}$ . Let $\gamma$ vary over the imaginary parts of the nontrivial zeros of $L(s, \pi)$ .

Theorem 1·3. Let $\pi \in \mathscr{A}$ . Let $\boldsymbol{\alpha} \in \mathbb{R}^n$ satisfy (1·12). If $h \in C^{n+2}(\mathbb{T}^n)$ , then

(1·16) \begin{equation} \sum_{|\gamma| \leq T} h(\gamma \boldsymbol{\alpha}) = N_{\pi}(T)\int_{\mathbb{T}^n} h(\boldsymbol{t}){d}\boldsymbol{t} + T \int_{\mathbb{T}^n} h(\boldsymbol{t}) g_{\pi, \boldsymbol{\alpha}}(\boldsymbol{t}){d}\boldsymbol{t}+o(T), \end{equation}

where ${d}\boldsymbol{t}$ is Lebesgue measure on $\mathbb{T}^n$ . The implied constant depends on $\pi$ , h, and $\boldsymbol{\alpha}$ .

Define the discrepancy

\begin{equation*} D_{\pi, \boldsymbol{\alpha}}(T) \;:\!=\; \sup_{\mathbb{B}\subseteq\mathbb{T}^n} \Big|\frac{1}{N_{\pi}(T)} \sum_{\substack{|\gamma|\leq T \\ \{\gamma \boldsymbol{\alpha}\} \in \mathbb{B}}}1-\textrm{vol}(\mathbb{B})\Big|.\end{equation*}

Our next result follows quickly from Theorem 1·3.

Corollary 1·4. Let $\pi\in\mathscr{A}$ . If $\boldsymbol{\alpha}\in\mathbb{R}^n$ satisfies (1·12), then

\begin{equation*} D_{\pi, \boldsymbol{\alpha}}(T) \geq \frac{T}{N_{\pi}(T)}\int_{\mathbb{T}^n}g_{\pi, \alpha}(\boldsymbol{t}){d}\boldsymbol{t}+o\Big(\frac{1}{\log T}\Big). \end{equation*}

When $n=1$ , we recover an unconditional analogue of (1·10) for all $\pi\in\mathscr{A}$ .This special case of Corollary 1·4 was proved in [ Reference Ford, Soundararajan and ZaharescuFSZ09 ] under the hypothesis of a zero density estimate of the form proved in Theorem 1·1.

1·2. Application to “zero races”

In a letter to Fuss, Chebyshev observed that the generalised Riemann hypothesis implies that primes $p\equiv 3\pmod{4}$ tend to be more numerous than primes $p\equiv 1\pmod{4}$ . Using the generalised Riemann hypothesis and other hypotheses, Rubinstein and Sarnak [ Reference Rubinstein and SarnakRS94 ] began a systematic study of “prime number races” in which they determine how often $\pi(x;4,3)>\pi(x;4,1)$ , where $\pi(x;q,a)$ equals $\#\{p\leq x\colon p\equiv a\pmod{q}\}$ . They proved that

\begin{equation*} \lim_{X\to\infty}\frac{1}{\log X}\int_{\substack{2\leq t\leq X \\ \pi(t;4,3)>\pi(t;4,1)}}\frac{dt}{t} = 0.9959\ldots\end{equation*}

Thus the “bias” toward primes of the form $4n+3$ is quite strong. The literature on prime number races which study such inequities is quite vast. See, for instance, the work of Fiorilli, Ford, Harper, Konyagin, Lamzouri and Martin [ Reference Fiorilli and MartinFM13, Reference Ford and KonyaginFK02, Reference Ford, Lamzouri and KonyaginFLK13, Reference Ford, Harper and LamzouriFHL19 ].

We use Theorem 1·3 to study inequities not between primes in different residue classes, but between zeros of different L-functions. Define

Let $h\in\mathscr{C}^{n+2}(\mathbb{T}^n)$ , and let $\boldsymbol{\alpha}$ satisfy (1·12). We consider two holomorphic cusp forms $f_1$ and $f_2$ with trivial nebentypus, where

\begin{equation*} f_j(z) = \sum_{n=1}^{\infty}\lambda_{f_j}(n)n^{\frac{k_j-1}{2}}e^{2\pi i n z}\in S_{k_j}^{\textrm{new}}(\Gamma_0(q_j)),\qquad j\in\{1,2\}\end{equation*}

have trivial nebentypus, even integral weights $k_j\geq 2$ and squarefree levels $q_j\geq 1$ . We assume that $f_j$ is normalised so that $\lambda_{f_j}(1)=1$ and that $f_j$ is an eigenfunction of all of the Hecke operators. We call such cusp forms newforms (see [ Reference OnoOno04 , section 2·5]). It is classical that there exists $\pi_j\in\mathscr{A}$ such that $L(s,f_j)=L(s,\pi_j)$ , so $\lambda_{f_j}(n) = \lambda_{\pi_j}(n)$ , $g_{f_j,\boldsymbol{\alpha}}(\boldsymbol{t})=g_{\pi_j,\boldsymbol{\alpha}}(\boldsymbol{t})$ , etc.

We say $f_1$ wins the $(\boldsymbol{\alpha}, h)$ -race (against $f_2$ ) if for all large T, we have

\begin{equation*} \sum_{|\gamma_1| \le T} h(\boldsymbol{\alpha} \gamma_1)> \sum_{|\gamma_2| \le T} h(\boldsymbol{\alpha} \gamma_2),\end{equation*}

where $\gamma_j$ runs over the imaginary parts of the nontrivial zeros of $L(s, f_j)$ . If neither $f_1$ nor $f_2$ wins the $(\boldsymbol{\alpha}, h)$ -race, we say the race is undecided. The results of [ Reference Liu and ZaharescuLZ21 ] suggest that the winner of a decided $(\boldsymbol{\alpha},h)$ -race is determined by the levels $q_1$ and $q_2$ and the behavior of $g_{f_1, \boldsymbol{\alpha}}$ and $g_{f_2, \boldsymbol{\alpha}}$ .

First, we show that if $q_1 = q_2$ and $h \in \mathscr{C}^{3}(\mathbb{T})$ , then the proportion of primes p such that $f_1$ wins the $({\log p}/{2\pi}, h)$ -race is ${1}/{2}$ .

Corollary 1·5. For $j=1,2$ , let $q_j$ be squarefree and $f_j \in S_{k_j}^{\textrm{new}} (\Gamma_0(q_j))$ be non-CM newforms. Suppose that $f_1\neq f_2\otimes\chi$ for all primitive Dirichlet characters $\chi$ . If $h \in \mathscr{C}^{3}(\mathbb{T})$ , then

\begin{equation*} \frac{\#\{ p \le X \colon \text{$f_1$ wins the $(\frac{\log p}{2\pi}, h)$-race}\}}{\#\{p\leq X\}} = \frac{1}{2}+O\Big(\frac{\sqrt{\log\log\log X}}{(\!\log\log X)^{1/4}}\Big). \end{equation*}

Next we look at the distribution of

\begin{equation*}H(f_1, f_2, h, \alpha) \;:\!=\; \lim_{T \to \infty} \frac{\sum_{|\gamma_1| < T} h( \alpha \gamma_1 ) - \sum_{|\gamma_2| < T} h(\alpha \gamma_2 ) }{T} - \frac{1}{\pi}\log\frac{q_{1}}{q_{2}}\int_{\mathbb{T}}h(t)dt\end{equation*}

as $\alpha$ varies over values of ${\log p}/{2\pi}$ for prime values of p. Given $I \subseteq [\!-\!2, 2]$ , let $\mu_{\textrm{ST}}(I)$ be the Sato–Tate measure $1/(2\pi)\int_{I}\sqrt{4-t^2}dt$ , and let $\mu_{ST,2}$ be the product measure defined on boxes $I_1\times I_2\subseteq [\!-\!2,2]\times[\!-\!2,2]$ by $\mu_{\textrm{ST},2}(I_1\times I_2)=\mu_{\textrm{ST}}(I_1)\mu_{\textrm{ST}}(I_2)$ . For $\mathcal{I} \subseteq [\!-\!4,4]$ , we define

(1·17) \begin{equation} \nu(\mathcal{I}) \;:\!=\; \mu_{ST,2} ( \{ (x,y) \in [\!-\!2,2]^2\;:\; x - y \in \mathcal{I} \} ).\end{equation}

For $h \in \mathscr{C}^3(\mathbb{T})$ , we set

\begin{equation*} k_h \;:\!=\; \int_0^1 h(t) \cos(2 \pi t) dt. \end{equation*}

In the following statement and throughout the paper, for any interval I and $\beta \in \mathbb{R}$ , we let $\beta I = \{ \beta x\;:\; x \in I \}$ .

Corollary 1·6. For $j=1,2$ , let $q_j$ be squarefree and $f_j \in S_{k_j}^{\textrm{new}} (\Gamma_0(q_j))$ be normalised holomorphic non-CM newforms with trivial nebentypus. Suppose that $f_1\neq f_2\otimes\chi$ for all primitive Dirichlet characters $\chi$ . Let

\begin{equation*}\varepsilon_X \;:\!=\; \frac{(\!\log\log\log X)^{1/4}}{(\!\log\log X)^{1/8}}. \end{equation*}

Assume $h \in \mathscr{C}^3(\mathbb{T})$ is such that $k_h \neq 0$ . Then for any interval $\mathcal{I}\subseteq[\!-\!4,4]$ , we have

\begin{equation*}\frac{\# \{ p \in [(1-\varepsilon_X)X, X]\colon \frac{\log{X}}{\sqrt{X}} H(f_1, f_2, h, \frac{\log{p}}{2 \pi}) \in \mathcal{I} \}}{\#\{p\in[(1-\varepsilon_X)X,X]\}}= \nu\Big(\frac{\pi}{2 k_h} \mathcal{I}\Big)+ O( \varepsilon_X )\end{equation*}

with an implied constant independent of $\mathcal{I}$ .

If $q_1 > q_2$ , then in contrast to Corollary 1·5, $f_1$ wins the $({\log p}/{2\pi},h)$ -race for all except finitely many primes p.

Corollary 1·7 shows that it is rare for $\pi_1$ to win an $(\alpha,h)$ race against $\pi_2$ if $q_2>q_1$ , but we can show this occurs infinitely often. Our result can be stated neatly in terms of local races rather than in terms of the $(\alpha, h)$ -races described above. For $t_0 \in \mathbb{T}$ , we say that $f_1$ wins the local $(\alpha, t_0)$ -race against $f_2$ if there exists a neighbourhood U of $t_0 \in \mathbb{T}$ such that the $(\alpha, h)$ -race is won by $f_1$ for all $h \in \mathscr{C}^{3}(\mathbb{T})$ which are supported on U.

In addition to Theorem 1·3, the proofs of these corollaries rely on a quantifiable understanding of the joint distribution of $\lambda_{f_1}(p)$ and $\lambda_{f_2}(p)$ as p varies over the primes. Such an understanding follows from the effective version of the Sato–Tate conjecture which counts the number of primes $p\leq X$ such that $(\lambda_{f_1}(p),\lambda_{f_2}(p))\in I_1\times I_2$ proved by the third author in [ Reference ThornerTho21 ] (see Theorem 2·1 below). One can prove analogues of Corollaries 1·5, 1·6 and 1·8 for Dirichlet L-functions by replacing the effective Sato–Tate estimates with results on primes in arithmetic progressions.