A prime number theorem in short intervals for dihedral Maass newforms

1.   Introduction

The classical prime number theorem asserts that

where $\Lambda(n)$ is the von Mangoldt function. One can deduce from this (cf. [1, §VII.2] for example) a prime number theorem for short intervals:

where the sum ranges over prime numbers, provided that $h$ is not too small. The proof of this result relies on the zero-free region

of the Riemann zeta-function $\zeta(s)$, and on the zero density estimate of the form

where $c, A, B$ are positive constants.

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{{GL}}_m(\mathbb{A}_ \mathbb{Q})$ with unitary central character, and $\tilde \pi$ be its contragredient. To generalize (1.1) to higher ranks, consider the Rankin-Selberg $L$-function

which has an analytic continuation and a functional equation (see Section 2.1 for more details). Here, $\alpha_{j, j'}^{\pi\times\tilde\pi}(p)$ depends on the Satake parameter $\alpha_{j }^{\pi }(p)$ and is equal to $\alpha_{j }^{\pi }(p)\overline{\alpha_{j' }^{\pi }(p)}$ at unramified primes. We define the generalized von Mangoldt function $\Lambda_{\pi\times\tilde\pi}(n)$ by the Dirichlet series identity

where $\Lambda_{\pi\times\tilde\pi}(p) = |\lambda_\pi(p)|^2\log p$ when $p$ does not divide the conductor $\mathfrak{q}_\pi$ of $\pi$. Here, $\lambda_\pi(p)$ is the eigenvalue of the Hecke operator at $p$.

It is well known that we have a prime number theorem for $L(s, \pi\times\tilde\pi)$ in the form

following from standard Rankin-Selberg theory and the Wiener-Ikehara Tauberian theorem (cf. [2, Lemma 5.2]). It is reasonable to expect that

when $h$ is not too small. For example, Motohashi ^[3] shows that, when $\pi$ is the cuspidal automorphic representation corresponding to a Hecke-Maass cusp form for $\operatorname{{SL}}(2, \mathbb{Z})$, there exist constants $c_0, \theta_0 > 0$ such that uniformly for $(\log x)^{-1/2}\leq\theta\leq\theta_0$,

for sufficiently large $x$. Unfortunately, a zero-free region for $L(s, \pi\times\tilde\pi)$ of the shape (1.2) does not yet exist for all $\pi$, so it seems impossible to prove a generalization of (1.1) except in special situations.

In this paper, we study the case when $\pi$ is the cuspidal automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$ corresponding to a dihedral Maass newform $\phi_k$ (see Section 2.2 for the definition). In Section 3.2 we will show the following asymptotic formula.

Theorem 1.1. Let $\pi$ be the automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$ corresponding to the dihedral form $\phi_k$. Then, for $x$ sufficiently large, we have

for $x^{13/16}\exp\big((\log x)^{4/5}\big)\leq h\leq x$.

The key ingredient of the proof is the factorization of the Rankin-Selberg $L$-function

where the $q$ is the level of $\phi_k$, and $\chi_q$ is the Dirichlet character defined by the Kronecker symbol $(\frac q\cdot)$. This factorization is why the zero-free region and the zero density estimate are both good enough to generalize (1.1): The $L$-function of $\phi_k$ coincides with that of a specific Hecke Grössencharacter, which has a zero-free region of Korobov-Vinogradov type according to ^[4]; and the zero density estimate of $L(s, \phi_k)$ is given by ^[5] and ^[6].

One can also write Theorem 1.1 as an estimate for sums of Hecke-Maass eigenvalues squared over primes in short intervals.

Corollary 1.2. Let $\lambda_k(p)$ be the Hecke eigenvalue of the dihedral form $\phi_k$. Then, for $x$ sufficiently large (in particular $x > q$), and for $x^{13/16}\exp\big((\log x)^{4/5}\big)\leq h\leq x$,

As an application, we give a lower bound for the proportion of primes in a short interval at which the Hecke eigenvalues of the dihedral form are greater than a given constant.

Corollary 1.3. Under the assumption of Corollary 1.2, for $x$ sufficiently large, there exists a prime $p\in (x, x+h]$ such that $|\lambda_k(p)|\geq 1$. In general, for any $0\leq\delta\leq 1$,

This paper is set up as follows. We begin by introducing our notation and defining the standard $L$-functions, Rankin-Selberg $L$-functions and dihedral Maass forms in Section 2. In Section 3.1 we recall some results on the zero density estimates, and in the rest of Section 3 we prove all the above theorem and corollaries.

2.   Priliminaries

2.1.   $L$-functions and the generalized von Mangoldt functions

We consider the following general setup. Let $F$ be a number field, $\mathcal{O}_F$ be its ring of integers and $\mathbb{A}_F$ the ring of adeles, and let $\pi = \otimes'_{ \mathfrak{p}}\pi_{ \mathfrak{p}}$ be a cuspidal automorphic representation of $\operatorname{{GL}}_m(\mathbb{A}_F)$. Assume that $\pi$ has a unitary central character. The local standard $L$-factor $L(s, \pi_{ \mathfrak{p}})$ at a prime ideal $\mathfrak{p}\subset \mathcal{O}_F$ is given by (cf. ^[7] or [8, §3.1] for example)

where $\alpha_1^{\pi}(\mathfrak{p}), \ldots, \alpha_m^{\pi}(\mathfrak{p})$ are the Satake parameters, and $\mathrm{N} \mathfrak{p} = \mathrm{N}_{F/ \mathbb{Q}} \mathfrak{p}:= \# \mathcal{O}_F/ \mathfrak{p}$ is the absolute norm of $\mathfrak{p}$. Denote by $\tilde \pi$ the contragredient of $\pi$. Then they have the same conductor $\mathfrak{q}_\pi = \mathfrak{q}_{\tilde\pi}$, and for each $\mathfrak{p}\nmid \mathfrak{q}_\pi$, we have $\alpha_j^{\tilde\pi}(\mathfrak{p}) = \overline{\alpha_j^{\pi}(\mathfrak{p})}$ (up to rearrangement of the parameters).

The global standard $L$-function $L(s, \pi)$ attached to $\pi$ is defined by

where the sum runs over the non-zero integral ideals of $F$, the product runs over the prime ideals of $F$, and $\mathrm{N} \mathfrak{n}:= \# \mathcal{O}_F/ \mathfrak{n}$ is the absolute norm of $\mathfrak{n}$. It has an analytic continuation apart from a simple pole at $s = 1$ when $\pi$ is trivial, and it satisfies the functional equation $\Lambda(s, \pi) = \varepsilon(\pi)\Lambda(1 - s, \tilde\pi)$ for all $s \in \mathbb{C}$ and for some complex number $\varepsilon(\pi)$ with $|\varepsilon(\pi)| = 1$. Here, the complete $L$-function $\Lambda(s, \pi)$ is defined by $L(s, \pi)$ times the infinite $L$-factors given by some Gamma functions, times a term depending on the conductor $\mathfrak{q}_\pi$. See [8, §3.1] for more details.

In particular, when $n = 2$ and $F = \mathbb{Q}$, each Hecke-Maass form $\phi$ corresponds to a cuspidal automorphic representation $\pi_\phi$ of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$. See, for instance, [9, §1.4] for the detailed description of the local representations $(\pi_\phi)_v$ and Satake parameters for each place $v$. By comparing the local $L$-factors, we notice that they have the same $L$-function $L(s, \phi) = L(s, \pi_\phi)$, where the Hecke $L$-function of $\phi$ (with Hecke eigenvalues $\lambda_{\phi}(n)$ and nebentypus $\chi$) is defined by (cf. [10, §5.11])

From now on, we consider the case when $F = \mathbb{Q}$. The Rankin-Selberg $L$-function $L(s, \pi\times\tilde\pi)$ can be defined by (cf. [11, §3.9] or [8, §3.2] for example)

where the local $L$-factors are given by

For example $\alpha_{j, j'}^{\pi\times\tilde\pi}(p) = \alpha_{j }^{\pi }(p)\overline{\alpha_{ j'}^{\pi }(p)}$ when $p\nmid \mathfrak{q}_\pi$. (See [12, §5.2] or [13, §A.1] for the definition of Satake parameters $\alpha_{j, j'}^{\pi\times\tilde\pi}(p)$ at ramified primes.) The above Dirichlet series and Euler product of $L(s, \pi\times\tilde\pi)$ both converge absolutely when $\mathrm{Re}(s) > 1$, and the Rankin-Selberg $L$-function $L(s, \pi\times\tilde\pi)$ also has an analytic continuation and functional equation, and has simple poles at $s = 0$ and $s = 1$.

The generalized von Mangoldt function $\Lambda_{\pi\times\tilde\pi}(n)$ is defined by the logarithmic derivative

By [14, Lemma a] we know that $\Lambda_{\pi\times\tilde\pi}(n)\geq 0$, and one can check that

In particular, we have

where the last identity holds when $p\nmid \mathfrak{q}_\pi$.

2.2.   Dihedral Maass forms

Let $K = \mathbb{Q}(\sqrt q)$ with fundamental discriminant $q$, where $q > 0$ is squarefree and $q\equiv 1\pmod 4$. Let $\chi_q$ be the quadratic character modulo $q$ associated to the extension $K/ \mathbb{Q}$ via class field theory, i.e., the Dirichlet character defined by the Kronecker symbol $(\frac q{\cdot})$.

For each integer $k\neq 0$, a dihedral Maass newform $\phi_k$ is the automorphic induction of a Hecke Grössencharacter $\Xi_k$ of $K$ for which $\Xi_k$ does not factor through the norm map $\mathrm{N}_{K/ \mathbb{Q}}$. We consider the characters $\Xi_k$ with conductor $\mathcal{O}_K$, which satisfy

for any principal ideal $(\alpha)\subset \mathcal{O}_K$, where $\alpha\mapsto\alpha'$ is the nontrivial automorphism in $\operatorname{Gal}(K/ \mathbb{Q})$. Let $K_\nu (z)$ be the modified Bessel function. Then the dihedral Maass newform $\phi_k$ is given by

where $e(\xi):= e^{2\pi i\xi}$, $\epsilon_q > 0$ is the fundamental unit of $K$, $\mathrm{N}\mathfrak{n}$ is the absolute norm of $\mathfrak{n}$, and $\rho_k(1)$ is the positive real number such that $\phi_k$ is $L^2$-normalized. It is well known (cf. ^[15], [16, §2], and [17, §1.1] for example) that $\phi_k$ is a Hecke-Maass cusp form on $\Gamma_0(q)$. It has level $\mathfrak{q}_{\phi_k} = q$, weight $0$, nebentypus $\chi_q$, and eigenvalue $\frac 14+ (\frac{ k\pi}{\log\epsilon_q})^2$.

The Hecke eigenvalue of $\phi_k$ at any $n$ can be calculated by $\lambda_{k}(n) = \sum_{ \mathrm{N}\mathfrak{n} = n}\Xi_k(\mathfrak{n})$. The Satake parameters $\alpha_1^{\phi_k}$, $\alpha_2^{\phi_k}$ of $\phi_k$ at a prime $p$ are related to the Hecke eigenvalue $\lambda_{k}(p)$ and nebentypus $\chi_q(p)$ via

Moreover, they are described in detail by the following fact.

Fact 2.1. ([18, §A.1]). The relationship between the Satake parameters $\alpha_1^{\phi_k}$, $\alpha_2^{\phi_k}$ of $\phi_k$ at a prime $p$ and the values of the Hecke Grössencharacter $\Xi_k$ on prime ideals $\mathfrak{p} \mid p \mathcal{O}_K$ is as follows:

(i) If $\chi_q(p) = 1$, i.e., $p$ splits in $K$ with $p \mathcal{O}_K = \mathfrak{p} \mathfrak{p}'$, then the Satake parameters are

(ii) If $\chi_q(p) = -1$, i.e., $p$ is inert in $K$ with $p \mathcal{O}_K = \mathfrak{p}$, then

(iii) If $\chi_q(p) = 0$, i.e., $p\mid q$, $p$ ramifies in $K$ with $p \mathcal{O}_K = \mathfrak{p}^2$, then

A direct corollary of the above fact is that the Hecke eigenvalue $\lambda_{k}(p)$ is real. One can also see this from that $\overline{\phi_k} = \phi_k$ is real by definition.

By comparing the local $L$-factors we have that: the Hecke $L$-function $L(s, \Xi_k)$ coincides with the classical $L$-function $L(s, \phi_k)$ for Maass forms; moreover, the Rankin-Selberg $L$-function satisfies the following factorization (cf. [10, §5.12] for example)

(in particular, we mention that $L_p(s, \phi_k\times\phi_k) = (1-p^{-s})^{-2}$ when $p\mid q$ by the the explicit descriptions in [12, §5.2]).

3.   Proof of the main theorem

3.1.   Zero density estimates

Let $\chi$ be any Dirichlet character modulo $q$ and $\phi$ be any Hecke-Maass newform. For $0\leq \sigma\leq 1$ and $T\geq 2$, we define

In other words, $N_*(\sigma, T)$ is the number of zeros of the corresponding $L$-function (or zeta-function) in the rectangle $|\mathrm{Im}\, \rho|\leq T$, $\sigma\leq \mathrm{Re}\, \rho < 1$. The GRH predicts that the non-trivial zeros of an $L$-function all lie on the critical line $\mathrm{Re}(s) = 1/2$, therefore $N_*(\sigma, T) = 0$ for $\sigma > 1/2$; and the zero-density conjecture states $N_*(\sigma, T)\ll T^{2(1-\sigma)}(\log T)^{B_*}$ for some $B_* > 0$.

In the proof of Theorem 1.1, we use the following uniform bounds for $N_*(\sigma, T)$ in the range $1/2\leq \sigma\leq 1$.

Lemma 3.1. Let $\chi$ be any (fixed) Dirichlet character modulo $q$ and $\phi$ be any (fixed) Hecke-Maass newform. For $1/2\leq \sigma\leq 1$, we have

Proof. These uniform bounds come from the following results of zero-density estimates: for $* = {\bf{1}}$, $\chi$ or $\phi$ we have $N_*(\sigma, T)\ll T^{A_*(\sigma)(1-\sigma)}(\log T)^{B_*}$ with

For any (fixed) Hecke-Maass newform $\phi$, one can follow the proofs in ^[5] and ^[6] to show that

□

3.2.   Proof of Theorem 1.1

For any cuspidal automorphic representation $\pi$ of $\operatorname{{GL}}_m(\mathbb{A}_ \mathbb{Q})$ whose central character is unitary, [8, (6.2)] shows the following explicit formula

where $\rho = \beta+i\gamma$ denotes the nontrivial zeros of $L(s, \pi \times \tilde{\pi})$. It follows that, for $2 \leq h \leq x$, we have

To estimate the sum over $\rho$, we observe that

Hence,

Furthermore,

where $N_{\pi \times \tilde{\pi}}(u, T)$ is the number of nontrivial zeros $\rho$ of $L(s, \pi \times \tilde{\pi})$ in the rectangle $|\mathrm{Im}\, \rho|\leq T$, $u\leq \mathrm{Re}\, \rho < 1$, and

By definition, we have that

and hence,

If $0\leq u < 1/2$, then we use the trivial bound (cf. [10, Theorem 5.8])

Assume that $N_{\pi \times \tilde{\pi}}(u, T)\ll T^{A(1-u)}(\log T)^B$ if $1/2\leq u < 1$, for some constants $A > 2$ and $B > 0$. In fact, when $\pi$ is the automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$ corresponding to the dihedral form $\phi_k$, one can take $A = 8/3$ and $B = 57$, by Lemma 3.1 and the factorization (2.4).

Recall that there exists a constant $c = c(q) > 0$ such that

for sufficiently large $T$ (cf. [22, §9.5] and [4, Theorem 2] for example, recall that $L(s, \phi_k) = L(s, \Xi_k)$ for the Hecke Grössencharacter $\Xi_k$). Assume $x\geq e\cdot T^A$ with $e$ the base of natural logarithms, and let $\theta(T) = c(\log T)^{-2/3}(\log\log T)^{-1/3}$. Again by the factorization (2.4) we have $N_{\pi \times \tilde{\pi}}(\sigma, T)\leq N_{{\bf{1}}}(\sigma, T)+N_{\chi_q}(\sigma, T)+N_{\phi_{2k}}(\sigma, T) = 0$ for $\sigma\geq 1- \theta(T)$. Then, we obtain the estimate for (3.2):

From this and (3.1) we have that when $\pi$ is the automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$ corresponding to $\phi_k$,

If we set

then we have that, for any

the remainder term is $O(\exp(-(\log x)^{\alpha-\frac 23 -\varepsilon}))$: more precisely,

At last, we take $A = 8/3$ and $\alpha = 4/5$ to complete the proof of Theorem 1.1.

3.3.   Proof of Corollary 1.2

Let $\pi$ be the automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$ corresponding to the dihedral form $\phi_k$. Recall (cf. (2.3) and Fact 2.1) that $|\alpha_{j }^{\pi }(p)|\leq 1$ and $|\lambda_k(p)|\leq 2$ for any prime $p$, which depend on the value $\Xi_k(\mathfrak{p})$ of the Hecke Grössencharacter $\Xi_k$ at the prime ideals $\mathfrak{p}\mid p \mathcal{O}_K$. Moreover, we have $|\alpha_{j, j'}^{\pi\times\tilde\pi}(p)|\leq 1$ because of the factorization (2.4). Therefore by the definition (2.2),

By Theorem 1.1 we have, for $x > q$ and $x^{13/16}\exp\big((\log x)^{4/5}\big)\leq h\leq x$,

Here $\lambda_{k}(p)$ is real via Fact 2.1. Note that in this short interval $p\in (x, x+h]$ we have that $\log x < \log p\leq\log 2x$, therefore $\log p = \log x+O(1)$. Hence,

3.4.   Proof of Corollary 1.3

We have

with

Combining (3.3) and (3.4), we get

i.e.,

This implies Corollary 1.3.

4.   Conclusions

In this paper, we study the prime number theorem in short intervals for $L$-functions of higher ranks. It is not easy to prove a prime number theorem in short intervals of the form (1.4) for the Rankin-Selberg $L$-function $L(s, \pi\times\tilde\pi)$ when $\pi$ is any cuspidal automorphic representation of $\operatorname{{GL}}_2(\mathbb{A}_ \mathbb{Q})$. But if $\pi$ corresponds to a dihedral Maass newform $\phi_k$, the $L$-function $L(s, \phi_k\times\phi_k)$ has a factorization (2.4) which leads to a nice zero-free region and a zero density estimate. Theorem 1.1 and Corollary 1.2 give two equivalent asymptotic formulas for the prime number theorem in short intervals for dihedral Maass newforms. As an application, in Corollary 1.3 we show a lower bound for the proportion of primes in a short interval at which the Hecke eigenvalues of the dihedral form are greater than a given constant.

Use of AI tools declaration

The author declares that he/she has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank Bingrong Huang and Yongxiao Lin for suggesting this problem and for helpful discussions. The author also thanks the anonymous referees for making helpful comments on an earlier version which led to improvement of the exposition. This research was completed while the author was supported by the National Key Research and Development Program of China (No. 2021YFA1000700).

Conflict of interest

The author declares no conflicts of interest.