Large sums of Hecke eigenvalues of holomorphic cusp forms

TL;DR

This paper studies the behavior of sums of Hecke eigenvalues of holomorphic cusp forms, establishing conditions under which these sums exhibit cancellations or large deviations, with implications for sign changes and automorphic form families.

Contribution

It provides new bounds and conditions for cancellations in Hecke eigenvalue sums, extending results analogous to character sums to automorphic forms and analyzing the range of $x$ relative to the weight $k$.

Findings

$S_f(x)=o(x ext{log} x)$ implies some eigenvalues are negative.

Under RH, $S_f(x)=o(x ext{log} x)$ holds for $x$ growing faster than $ ext{log}k/ ext{log} ext{log}k$.

Existence of many forms with large sums $S_f(x) ext{gg} x ext{log} x$ when $x=( ext{log}k)^A$.

Abstract

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\sum_{n\leq x} \lambda_f(n)$. We first show that $S_f(x)=o(x\log x)$ implies that $\lambda_f(n)<0$ for some $n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms $f$ of large weight $k$, for which $S_f(x)\gg_A x\log x$, when $x=(\log k)^A.$ Our results are $GL_2$ analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

Equations

$$|{\mathcal{H}}_k|=\frac{k}{12}+O\left(k^{2/3}\right).$$

$$f(z)=\sum _{n=1}^{\infty }{\lambda }_f(n)n^{(k-1)/2}e(nz),\text{ for }\text{Im}(z)>0,$$

$${\lambda }_f(m){\lambda }_f(n)=\sum _{d|(m,n)}{\lambda }_f\left(\frac{mn}{d^2}\right),$$

$$\left|{\lambda }_f(n)\right|\le \tau (n),$$

$$n_f\ll k^{3/4}.$$

$$S_f(x):=\sum _{n\le x}{\lambda }_f(n),$$

$$\big{|}S_{f}(x)\big{|}\leq\sum_{n\leq x}\tau(n)=(1+o(1))x\log x.$$

$$S_f(x)=o(x\log x)(\text{as }x,k\to \infty ),$$

$$\sum _{n\le x}{\lambda }_f(n)\ge c_{0}x\log x,$$

$$L(s,f)=\sum _{n=1}^{\infty }\frac{{\lambda }_f(n)}{n^s}=\prod _p{\left(1-\frac{e^{i{\theta }_f(p)}}{p}\right)}^{-1}\prod _p{\left(1-\frac{e^{-i{\theta }_f(p)}}{p}\right)}^{-1},\text{ for }\text{Re}(s)>1,$$

$$S_f(x)\ll x^{1/2+\epsilon }\cdot k^{1/2+\epsilon },$$

$$S_f(x)\ll x^{1/2+\epsilon }\exp \left(c_1\frac{\log k}{\log \log k}\right),$$

$$x\ge \exp \left(c_2\frac{\log k}{\log \log k}\right)$$

$$\sum _{n\le x}{\lambda }_f(n)=o(x\log x).$$

$$\sum _{n\le x}{\lambda }_f(n)=\sum _{\begin{array}{c}n\le x\\ P(n)\le y\end{array}}{\lambda }_f(n)+O\left(\frac{(\log k)(\log y{)}^4}{\sqrt{y}}x\log x\right).$$

$$\Psi (x,y;g):=\sum _{\begin{array}{c}n\le x\\ P(n)\le y\end{array}}g(n).$$

$$\Psi (x,y;\tau )\sim {\rho }_2(u)\cdot x\log y,\text{ where }u:=\frac{\log x}{\log y},$$

$$u{\rho }_2^{\prime }(u)={\rho }_2(u)-2{\rho }_2(u-1),$$

$$\Psi (x,y;{\lambda }_f)\le \Psi (x,y;\tau )\ll u^{-u(1+o(1))}x\log x,$$

$$\Psi (x,y;\tau )\ll e^{-u/2}x\log x.$$

$$S_f(x)\gg x\log x.$$

$$\sum _{n\le x}{\lambda }_f(n){\gg }_{A}x\log x.$$

$$\left|\sum _{n\le x}{\lambda }_f(n)\right|\ge \Psi (x,y;\tau )\left(1+O\left(\frac{1}{\sqrt{\log \log k}}\right)\right).$$

$$\mathbb{X}(p^a)=\text{tr}\left({\text{Sym}}^a\left(g_p^{♮}\right)\right)$$

$$\mathbb{X}(n)=\mathbb{X}(p_1^{a_1})\cdots \mathbb{X}(p_{\ell }{)}^{a_{\ell }}$$

$${\lambda }_f(p^b)=\frac{\sin ((b+1){\theta }_f(p))}{\sin {\theta }_f(p)},$$

$$h_x(p)=\{\begin{array}{ll}\alpha \left(\frac{\log p}{\log x}\right)& \text{ if }p\le x,\\ 0& \text{ otherwise.}\end{array}$$

$$\sum _{n\le x}{\lambda }_f(n)\ge \sum _{n\le x}{h}_x(n).$$

$$\sum _{n\le x}{\lambda }_f(n)\ge \text{sideset}{}^{♭}{\sum }_{n\le x}{\lambda }_f(n),$$

$$0\le {\lambda }_f(p^j)=\frac{\sin ((j+1){\theta }_f(p))}{\sin {\theta }_f(p)}.$$

Full text

The author is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

1. Introduction

Let $k$ be a positive even integer, and denote by $\mathcal{H}_{k}$ the set of Hecke cusp forms of weight $k$ for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Then, $\mathcal{H}_{k}$ is an orthonormal basis for the space of holomorphic cusp forms of weight $k$ for $\Gamma$ and we have

[TABLE]

Given $f\in\mathcal{H}_{k}$, its Fourier expansion can be written in the form

[TABLE]

where $e(z)=e^{2\pi iz}.$ The $\lambda_{f}(n)$ are the normalized eigenvalues of the Hecke operators $T_{n}$, and satisfy the well-known Hecke relations:

[TABLE]

for all $m,n\geq 1$. In particular, $\lambda_{f}$ is a real-valued multiplicative function of $n$. Moreover, it also satisfies the following deep bound due to Deligne

[TABLE]

where $\tau$ is the divisor function. These facts are standard and may be found for example in Chapter 14 of [8].

In [9], Kowalski, Lau, Soundararajan and Wu studied the signs of the sequence $\lambda_{f}(n)$. Their results show a strong analogy between these signs and the values of quadratic Dirichlet characters, and especially between the first negative Fourier coefficient and the problem of the least quadratic non-residue, which has a long history in analytic number theory. Let $n_{f}$ be the smallest positive integer $n$ such that $\lambda_{f}(n)<0$. The best known bound for $n_{f}$ is due to Matomäki [12], who improved the authors of [9] by showing that

[TABLE]

This is probably far from the truth, since it is known that $n_{f}\ll(\log k)^{2}$ under the assumption of the generalized Riemann hypothesis (GRH). In the other direction, Theorem 3 of [9] shows that $n_{f}\gg\sqrt{\log k}$ for many Hecke cusp forms $f$ of weight $k$. A folklore conjecture asserts that the correct order of magnitude for the maximal values of $n_{f}$ should be $(\log k)^{1+o(1)}$.

In this paper, we explore $GL_{2}$ analogues of certain classical problems concerning short character sums and the least quadratic non-residue. More precisely, we investigate the size of the short sum of Hecke eigenvalues

[TABLE]

and its relation to the first negative Fourier coefficient of $f$. Our results are inspired by the work of Granville and Soundararajan [4] on character sums. In particular, Corollaries 1.2 and 1.4 below can be regarded as $GL_{2}$ analogues of Corollary A of [4].

Using Deligne’s bound (1.2), one obtains the “trivial” bound

[TABLE]

Our first result shows that if $S_{f}(x)$ is substantially smaller than this bound, namely that

[TABLE]

then we must have $n_{f}\leq x$. The proof relies on an argument of Kowalski, Lau, Soundararajan and Wu [9], together with a nice result of Hildebrand [6] concerning quantitative lower bounds for mean values of non-negative multiplicative functions.

Theorem 1.1**.**

Let $f\in\mathcal{H}_{k}$. Let $x\geq 2$ and assume that $\lambda_{f}(n)\geq 0$ for all $n\leq x$. Then, we have

[TABLE]

for some absolute constant $c_{0}>0$.

Let $f\in\mathcal{H}_{k}$. The $L$-function attached to $f$ is defined by

[TABLE]

where $\theta_{f}(p)\in[0,\pi]$. It is known that $L(s,f)$ extends analytically to the entire complex plane, and satisfies a functional equation that relates $L(s,f)$ to $L(1-s,f)$ (see for example Section 5.11 of [8]). A standard application of Perron’s formula together with the convexity bound for $L(s,f)$ imply that

[TABLE]

and hence one has $S_{f}(x)=o(x\log x)$ in the range $x\geq k^{1+\varepsilon}.$ This range can be improved to $x\geq k^{1-\delta}$, for some $\delta>0$, by using subconvexity bounds for $L(s,f)$ (see for example [13]). Furthermore, assuming GRH for $L(s,f)$ one has the much stronger bound

[TABLE]

for some absolute constant $c_{1}>0$. This shows that (1.3) is valid in the larger range

[TABLE]

for some constant $c_{2}>0$, conditionally on the GRH. Exploiting an idea of Montgomery and Vaughan [14], we substantially improve this range under the assumption of GRH.

Corollary 1.2**.**

Let $f\in\mathcal{H}_{k}$, and assume GRH for $L(s,f)$. In the range $\log x/\log\log k\to\infty$, we have

[TABLE]

We shall deduce this result from the following theorem, which shows that under GRH, we can approximate $S_{f}(x)$ by the corresponding sum of $\lambda_{f}(n)$ over friable (or smooth) numbers $n$, which are positive integers having only small prime factors. A positive integer $n$ is said to be $y$-friable if $P(n)\leq y$, where $P(n)$ denotes the largest prime factor of $n$, with the standard convention $P(1)=1$.

Theorem 1.3**.**

Let $f\in\mathcal{H}_{k}$, and assume GRH for $L(s,f)$. Then, for all real numbers $x,y$ such that $(\log k)^{2}(\log\log k)^{8}\leq y\leq x\leq k$ we have

[TABLE]

For an arithmetic function $g$, we define

[TABLE]

The asymptotic behaviour of $\Psi(x,y;g)$ was investigated for a large class of multiplicative functions $g$ by several authors, and notably by Tenenbaum and Wu [18]. When $g$ is the divisor function $\tau$, de Bruijn and van Lint [1] proved that there exists a differentiable function $\rho_{2}:[0,\infty)\to\mathbb{R}$ such that

[TABLE]

in the range $u\ll 1$. The function $\rho_{2}$ is defined by the differential-difference equation

[TABLE]

subject to the initial condition $\rho_{2}(u)=u$ for $0\leq u\leq 1$. It is known that $\rho_{2}(u)>0$ for any $u>0$ and that $\rho_{2}(u)=u^{-u(1+o(1))}$ for large $u$ (see for example [5]). In fact, $\rho_{2}$ is the square convolution of the standard Dickman-de Bruijn function $\rho$, which appears in the asymptotic formula for the counting function of friable integers. The range of validity of the asymptotic formula (1.7) was improved to $u\leq\exp\left((\log y)^{3/5-\varepsilon}\right)$ by Smida [16], and hence in this range we have

[TABLE]

by (1.2). For our purposes, it is enough to use the following weaker bound that holds uniformly for $10\leq y\leq x$ (see Lemma 4.3 below)

[TABLE]

Combining this bound with Theorem 1.3 imply Corollary 1.2.

We now investigate the largest range of $x$ (in terms of $k$) for which one has

[TABLE]

Recall that $n_{f}\gg\sqrt{\log k}$ for many Hecke cusp forms $f$ of weight $k$ by Theorem 3 of [9]. In view of Theorem 1.1, this shows that (1.10) is valid for such $f$ with $x=\sqrt{\log k}$. On the other hand, since $n_{f}\ll(\log k)^{2}$ on GRH, one might guess that (1.10) does not hold in the range $x\gg(\log k)^{2+\varepsilon}$. We prove that this is not the case, by showing that for any $A>1$, there are many Hecke cusp forms $f$ of weight $k$ such that (1.10) holds for $x=(\log k)^{A}$. This shows that the range of Corollary 1.2 is best possible, and that conditionally on GRH the converse of Theorem 1.1 does not hold.

Corollary 1.4**.**

Let $k$ be a large even integer. Let $A>1$ be fixed, and $x=(\log k)^{A}$. There are at least $k^{1-1/\log\log k}$ Hecke cusp forms $f\in\mathcal{H}_{k}$ such that

[TABLE]

We shall deduce this result from the following theorem.

Theorem 1.5**.**

Let $k$ be a large even integer. Let $A>1$ be fixed, $y=\log k/\log\log k$ and $x=(\log k)^{A}$. There are at least $k^{1-1/\log\log k}$ Hecke cusp forms $f\in\mathcal{H}_{k}$ such that

[TABLE]

The key idea in the proof of Theorem 1.5 is to compare large moments of $S_{f}(x)$ (as $f$ varies in $\mathcal{H}_{k}$) with those of a corresponding probabilistic random model. This model was introduced by Cogdell and Michel in [2] to study the complex moments of symmetric power $L$-functions at $s=1$, and was subsequently used by various authors (see for example [10] and [11]) to explore similar problems. To describe this probabilistic model we consider the compact group $G=SU(2)$ endowed with its natural Haar measure $\mu_{G}$; we then let $G^{\natural}$ be the set of conjugacy classes of $G$ endowed with the Sato-Tate measure $\mu_{st}$ (i.e. the direct image of $\mu_{G}$ by the canonical projection). Let $\{g^{\natural}_{p}\}_{p\text{ prime }}$ be a sequence of independent random variables, with values in $G^{\natural}$ and distributed according to the measure $\mu_{st}$. We construct the sequence of random variables $\{\mathbb{X}(n)\}_{n\geq 1}$ by first defining

[TABLE]

for a prime $p$ and a positive integer $a$, where $\text{Sym}^{a}$ is the symmetric $a$-th power representation of the standard representation of $GL_{2}$. We then extend the $\mathbb{X}(p^{a})$ multiplicatively by letting $\mathbb{X}(1)=1$ and

[TABLE]

if the prime factorization of $n$ is $n=p_{1}^{a_{1}}\cdots p_{\ell}^{a_{\ell}}.$ We shall explore this probabilistic model and the motivation behind it in details in Section 3. Using the Petersson trace formula (see Lemma 3.1 below), we show that in a certain range of $x$, large (weighted) moments of $S_{f}(x)$ are very close to those of the sum of random variables $\sum_{n\leq x}\mathbb{X}(n)$. We then estimate the moments of this sum by first restricting the random variables $\mathbb{X}(n)$ to those indexed by $y$-friable integers $n$, and then controlling these by restricting the range of the random variables $\mathbb{X}(p)$ for the primes $p\leq y$.

Our approach is flexible and could be further generalized to obtain similar results for other families of automorphic forms. In particular, our results hold mutatis mutandis for primitive Hecke cusp forms of weight $2$ and prime level $q$ (in the level aspect), with the extra condition that $x<q$ in Theorem 1.1. One should also obtain the analogues of Theorems 1.3 and 1.5 for Fourier coefficients of the symmetric square and other symmetric power $L$-functions attached to primitive Hecke cusp forms, assuming their automorphy.

2. The size of $S_{f}(x)$ and the first negative Hecke eigenvalue: Proof of Theorem 1.1

Let $p$ be a prime number. It follows from (1.4) that $\lambda_{f}(p)=2\cos\theta_{f}(p)$ and more generally we have

[TABLE]

for any integer $b\geq 0$, by the Hecke relations (1.1).

Let $\alpha:[0,1]\to[-2,2]$ be defined by $\alpha(0)=2$ and $\alpha(t)=2\cos(\pi/(m+1))$ if $1/(m+1)<t\leq 1/m$, for $m\in\mathbb{N}$. For $x\geq 2$, let $h_{x}$ be the multiplicative function supported on square-free numbers and defined on the primes by

[TABLE]

By exploiting the Hecke relations (1.1), we obtain the following lemma which is essentially proved in [9].

Lemma 2.1**.**

Let $f\in\mathcal{H}_{k}$. Let $x\geq 2$ be such that $\lambda_{f}(n)\geq 0$ for all $n\leq x$. Then, we have

[TABLE]

Proof.

By our assumption we have

[TABLE]

where $\sideset{}{{}^{\flat}}{\sum}$ restricts the summation to squarefree integers. Since $h_{x}(n)\geq 0$ for all squarefree $n$, it thus suffices to show that $\lambda_{f}(p)\geq h_{x}(p)$ for all primes $p\leq x$. Let $p\leq x$ be a prime number, and $m\geq 1$ be such that $x^{1/(m+1)}<p\leq x^{1/m}$. Then, for all integers $1\leq j\leq m$ we have

[TABLE]

This implies $0\leq\theta_{f}(p)\leq\pi/(m+1)$ and hence that

[TABLE]

as desired. ∎

In order to complete the proof of Theorem 1.1, we need to obtain a lower bound for $\sum_{n\leq x}h_{x}(n)$. We prove the following result.

Proposition 2.2**.**

There is an absolute constant $c_{0}>0$ such that for all large $x$ we have

[TABLE]

Combining this result with Lemma 2.1 imply Theorem 1.1. In order to prove Proposition 2.2, we shall use the following theorem of Hildebrand [6] which provides quantitative lower bounds for mean values of certain non-negative multiplicative functions.

Theorem 2.3** (Theorem 2 of [6]).**

Let $2\leq z\leq x$ be real numbers. Let $g$ be a multiplicative function supported on squarefree numbers, such that $0\leq g(p)\leq K$ for some constant $K\geq 1$ and all primes $p$. Then, we have

[TABLE]

where $\gamma$ is the Euler-Mascheroni constant, $y^{+}=\max\{y,0\}$, $\beta>0$ is an absolute constant, and $\sigma(u)$ is a continuously differentiable function of $u\geq 1$ that satisfies $\sigma(u)\gg u^{-u}$. Furthermore, the implicit constants in the $O$-terms depend on $K$ only.

We also need the following lemma.

Lemma 2.4**.**

Let $x$ be large. Then, we have

[TABLE]

Proof.

First, note that

[TABLE]

Let $M<\sqrt{\log x}$ be a large positive integer to be chosen later. Then, we have

[TABLE]

Furthermore, we have

[TABLE]

Choosing $M=[\log\log x]$, and inserting the estimates (2.2) and (2.3) in (2.1) completes the proof. ∎

Proof of Proposition 2.2.

Note that $h_{x}(p)<1$ if and only if $p>x^{1/2}$. Therefore, for all $z\leq x^{1/2}$ we have

[TABLE]

Thus, choosing $K=z=2$ in Theorem 2.3 we obtain that

[TABLE]

The result follows from Lemma 2.4. ∎

3. Large sums of Hecke eigenvalues : proofs of Theorem 1.5 and Corollary 1.4

In order to prove Theorem 1.5, we shall compute the moments of $S_{f}(x)$ as $f$ varies in $\mathcal{H}_{k}$. When so doing, we shall use the harmonic weights that arise naturally in the Petersson trace formula (see Lemma 3.1 below). The harmonic weight of $f\in\mathcal{H}_{k}$ is defined by

[TABLE]

where $\langle f,f\rangle$ is the Petersson inner product, and $L(s,\text{Sym}^{2}f)$ is the symmetric square $L$-function of $f$. Given a sequence $(\alpha_{f})_{f\in\mathcal{H}_{k}}$, its harmonic average is defined as the sum

[TABLE]

and if $S\subset\mathcal{H}_{k}$ we will let $|S|_{h}$ denote the harmonic measure of $S$, that is

[TABLE]

Moreover, the classical estimate

[TABLE]

together with the bounds of Goldfeld, Hoffstein and Liemann (see the Appendix of [7])

[TABLE]

show that the harmonic weight $\omega_{f}$ is close to the natural weight $1/|\mathcal{H}_{k}|$ (since $|\mathcal{H}_{k}|\asymp k$), and it defines asymptotically a probability measure on $\mathcal{H}_{k}$.

We shall use the following consequence of the Petersson trace formula which follows from Lemma 2.1 of [15].

Lemma 3.1**.**

Let $k$ be a large even integer, and $n$ be a positive integer such that $n\leq k^{2}/10^{4}$. Then, we have

[TABLE]

where $\delta(n)=1$ if $n=1$, and is [math] otherwise.

Proof.

It follows from Lemma Lemma 2.1 of [15] that

[TABLE]

The result follows from combining this estimate with (3.1). ∎

The formula (3.3) can be interpreted as follows: Recall that $G^{\natural}$ is the set of conjugacy classes of $G=SU(2)$ endowed with the Sato-Tate measure $\mu_{st}$ (the direct image of the Haar measure $\mu_{G}$ by the canonical projection). Let $n>1$ and $n=p_{1}^{a_{1}}\cdots p_{\ell}^{a_{\ell}}$ be its prime factorization. Then we have the identity

[TABLE]

where

[TABLE]

Fix now the primes $p_{1},\dots,p_{\ell}$. By the identity (3.4) together with the Peter-Weyl Theorem and Weyl’s equidistribution criterion, the estimate (3.3) applied to integers $n$ divisible only by the primes in $\{p_{1},\dots,p_{\ell}\}$ yields the equidistribution of the $\ell$-tuple of conjugacy classes $\{g^{\natural}_{f}(p_{1}),\dots,g^{\natural}_{f}(p_{\ell})\}_{f\in\mathcal{H}_{k}}$ (appropriately weighted by $\omega_{f}$) into the product of $\ell$ copies of $G^{\natural}$, as $k\to\infty$. Based on this equidistribution result, we construct a probabilistic random model for the Hecke eigenvalues $\lambda_{f}(n)$ as follows: let $\{g^{\natural}_{p}\}_{p\text{ prime }}$ be a sequence of independent random variables, with values in $G^{\natural}$ and distributed according to the measure $\mu_{st}$. We define $\mathbb{X}(1)=1$ and for $n>1$

[TABLE]

if $n=p_{1}^{a_{1}}\cdots p_{\ell}^{a_{\ell}}$ is the prime factorization of $n$. Furthermore, one can easily check that the $\mathbb{X}(n)$ satisfy the Hecke relations (1.1), namely that

[TABLE]

We prove the following lemma.

Lemma 3.2**.**

Let $n$ be a positive integer. Then we have

[TABLE]

Proof.

Let $n>1$, and write the prime factorization of $n$ as $n=p_{1}^{a_{1}}\cdots p_{\ell}^{a_{\ell}}$. First, by the independence of the random variables $g^{\natural}_{p}$ for different primes $p$, we have

[TABLE]

By Weyl’s integration formula, the map

[TABLE]

identifies $G^{\natural}$ with the interval $[0,\pi]$ and $\mu_{st}$ with the distribution $d\mu_{st}(t):=\frac{2}{\pi}\sin^{2}(t)dt$. Furthermore, note that

[TABLE]

and hence

[TABLE]

Therefore, for a prime $p$ and a positive integer $a$ we obtain

[TABLE]

since the functions $\{Y_{n}\}_{n\geq 0}$, defined by

[TABLE]

form an orthonormal basis of $L^{2}([0,\pi],d\mu_{st})$. This completes the proof.∎

Using Lemmas 3.1 and 3.2 we prove that in a certain range of $x$, the harmonic moments of $S_{f}(x)$ (as $f$ varies in $\mathcal{H}_{k}$) are very close to the moments of the sum of random variables $\sum_{n\leq x}\mathbb{X}(n)$.

Proposition 3.3**.**

Let $k$ be a large even integer. Let $x\geq 2$ and $\ell$ be a positive integer such that $x^{6\ell}\leq k.$ Then, we have

[TABLE]

In order to prove this proposition, we need to understand the combinatorics of the Hecke relations (1.1). These relations can be written as

[TABLE]

where $b_{m}(n_{1},n_{2})=1$ if $m=n_{1}n_{2}/d^{2}$ for some $d|(n_{1},n_{2})$, and equals [math] otherwise. More generally, one can write

[TABLE]

for some integers $b_{m}(n_{1},\dots,n_{r})$. These coefficients have a nice interpretation in terms of the representation theory of $G=SU(2)$. The irreducible characters of $G$ are

[TABLE]

for $a\geq 0$. Hence, for $n=p_{1}^{a_{1}}\cdots p_{\ell}^{a_{\ell}}$, the character

[TABLE]

is an irreducible character of the product of $\ell$ copies of $G$, and the formula

[TABLE]

is the decomposition formula for the product of the $r$ characters $\chi_{n_{1}},\dots,\chi_{n_{r}}$ in terms of the irreducibles $\chi_{m}$. In particular, the coefficients $b_{m}(n_{1},\dots,n_{r})$ are non-negative, and we also have

[TABLE]

Moreover, one can easily prove (either by induction on $r$ or by exploiting the representation theory of $SU(2)$) that

[TABLE]

Lemma 3.4**.**

Let $g$ be a real-valued arithmetic function. For all $x\geq 2$ and positive integers $\ell$ we have

[TABLE]

Proof.

We have

[TABLE]

Moreover, it follows from (3.6) that

[TABLE]

by Lemma 3.2. This completes the proof. ∎

We deduce the following corollary.

Corollary 3.5**.**

Let $g$ and $h$ be arithmetic functions such that $g(n)\geq h(n)\geq 0$ for all $n\geq 1$. Then we have

[TABLE]

We are now ready to prove Proposition 3.3.

Proof of Proposition 3.3.

By (3.5) we obtain

[TABLE]

Therefore, by Lemma 3.1 we get

[TABLE]

Now, using that $b_{m}(n_{1},\dots,n_{2\ell})\leq\tau(n_{1})\cdots\tau(n_{2\ell})$ we deduce that the error term above is

[TABLE]

using the bound $\tau(n_{1}\cdots n_{2\ell})\ll_{\varepsilon}(n_{1}\cdots n_{2\ell})^{\varepsilon}\leq x^{2\ell\varepsilon}$ together with the estimate $\sum_{n\leq x}\tau(n)\ll x\log x$. Appealing to Lemma 3.4 completes the proof. ∎

To complete the proof of Theorem 1.5 we need to derive lower bounds for the moments of $\sum_{n\leq x}\mathbb{X}(n)$. We establish the following proposition.

Proposition 3.6**.**

Let $\ell\geq 2$ be an integer. Then, for all real numbers $2\leq y\leq x$ we have

[TABLE]

Proof.

First, by Corollary 3.5 with $g(n)=1$ and $h(n)$ being the characteristic function of the $y$-friable numbers, we get

[TABLE]

For a prime $p$, write

[TABLE]

where $\theta_{p}$ is a random variable taking values in $[0,\pi]$ and distributed according to the Sato-Tate distribution $d\mu_{st}(t):=\frac{2}{\pi}\sin^{2}(t)dt$ . Let $\mathcal{A}(\mathbb{X})$ be the event corresponding to

[TABLE]

By the independence of the $\theta_{p}$ for different primes $p$, we deduce that the probability of $\mathcal{A}(\mathbb{X})$ is

[TABLE]

for some positive constant $c$. On the other hand, one can see that for any prime $p\leq y$ and all outcomes in $\mathcal{A}(\mathbb{X})$, we have

[TABLE]

Therefore, if $n\leq x$ and $P(n)\leq y$ then for all outcomes in $\mathcal{A}(\mathbb{X})$ we have

[TABLE]

where $\omega(n)$ is the number of distinct prime factors of $n$, which satisfies $\omega(n)\ll\log x$. Thus, we deduce that

[TABLE]

as desired. ∎

We finish this section by proving Theorem 1.5, and deducing Corollary 1.4.

Proof of Theorem 1.5.

Let $\ell=[\log k/(6\log x)]$. Then, it follows from Proposition 3.3 and Proposition 3.6 that

[TABLE]

Therefore, in view of (3.1) and (3.2) we obtain

[TABLE]

Let $\mathcal{B}$ be the set of Hecke cusp forms $f\in\mathcal{H}_{k}$ such that

[TABLE]

Since $|\mathcal{H}_{k}|\asymp k$ we obtain

[TABLE]

Combining this bound with (3.9) we get

[TABLE]

On the other hand, we have

[TABLE]

Moreover, by (1.7) we have

[TABLE]

Hence, we derive from (3.10) that

[TABLE]

which completes the proof. ∎

Proof of Corollary 1.4.

The result follows from Theorem 1.5 together with Eq. (3.11). ∎

4. Cancellations under GRH: proofs of Theorem 1.3 and Corollary 1.2

Let $f\in\mathcal{H}_{k}$. For $\textup{Re}(s)>1$ we have

[TABLE]

where $b_{f}(n)=(e^{i\theta_{f}(p)})^{a}+(e^{-i\theta_{f}(p)})^{a}$ if $n=p^{a}$ for some prime $p$, and equals [math] otherwise. For $y\geq 1$ we define

[TABLE]

In order to approximate $S_{f}(x)$ by $\Psi(x,y;\lambda_{f})$, we shall prove that conditionally on GRH, $\log L(s,f)$ is very well approximated by $\log L_{y}(s,f)$ for $\textup{Re}(s)\geq 1$. This will be the key ingredient in the proof of Theorem 1.3.

Lemma 4.1**.**

Let $f\in\mathcal{H}_{k}$ and assume GRH for $L(s,f)$. Let $2\leq y\leq k$, and $s=\sigma+it$ with $\sigma\geq 1$ and $|t|\leq 2k$. Then, we have

[TABLE]

To prove this result we need the following standard bound.

Lemma 4.2**.**

Let $f\in\mathcal{H}_{k}$. Let $s=\sigma+it$ with $1/2<\sigma\leq 3/2$ and $|t|\leq 3k$. Let $1/2\leq\sigma_{0}<\sigma$, and suppose that there are no zeros of $L(z,f)$ inside the rectangle $\{z:\sigma_{0}\leq\textup{Re}(z)\leq 1,|\textup{Im}(z)-t|\leq 3\}$. Then, we have

[TABLE]

Proof.

Consider the circles with centre $2+it$ and radii $r=2-\sigma$ and $R=2-\sigma_{0}$, so that the smaller circle passes through $s$. By our assumption, $\log L(z,f)$ is analytic inside the larger circle. For a point $z$ on the larger circle, it follows from the standard convexity bound for $L(s,f)$ that $\textup{Re}\log L(z,f)\ll\log k$. Finally, using the Borel-Caratheodory theorem we obtain

[TABLE]

∎

Proof of Lemma 4.1.

Let $c_{1}=1-\sigma+1/\log y$. Then it follows from Perron’s formula (see [3]) that

[TABLE]

by a standard estimation of the error term. We now move the contour to the line $\textup{Re}(s)=c_{2}$ where $c_{2}=1/2-\sigma+1/\log y$. By our assumption, we only encounter a simple pole at $z=0$ that leaves a residue of $\log L(s,f)$. Furthermore, it follows from Lemma 4.2 with $\sigma_{0}=1/2$ that

[TABLE]

uniformly for $z$ with $\textup{Re}(z)\geq c_{2}$ and $|\textup{Im}(z)|\leq y$. Therefore, we deduce that

[TABLE]

where

[TABLE]

The result follows upon noting that

[TABLE]

∎

We now prove Theorem 1.3.

Proof of Theorem 1.3.

Without loss of generality assume that $x\in\mathbb{Z}+1/2$. Let $c=1+1/\log x$. By Perron’s formula together with (1.2) we have

[TABLE]

The error term above is

[TABLE]

Similarly, we have

[TABLE]

Define

[TABLE]

Then, combining the above estimates we get

[TABLE]

Moreover, using Lemma 4.1 we obtain

[TABLE]

for all $s$ with $\textup{Re}(s)=c$ and $|\textup{Im}(s)|\leq x$. Furthermore, note that for $\textup{Re}(s)=c$ we have

[TABLE]

since $\lambda_{f}(p)\leq 2$. Combining these estimates, we deduce that

[TABLE]

which completes the proof.

∎

In order to deduce Corollary 1.2, we need to prove the bound (1.9), which shows that $\Psi(x,y;\tau)=o(x\log x)$ when $u=\log x/\log y\to\infty$.

Lemma 4.3**.**

Let $10\leq y\leq x$ be real numbers. Then we have

[TABLE]

Proof.

Let $\beta=2/(3\log y)$. Then, observe that

[TABLE]

Let

[TABLE]

Then $g$ is multiplicative, and for all primes $p\leq y$ we have $g(p^{a})=(a+1)p^{a\beta}\ll(1.9)^{a}$. Therefore, by Corollary 3.5.1 of [17] we obtain

[TABLE]

The result follows upon noting that $x^{3/4}\ll xe^{-u/2}$ for $y\geq 10$ and

[TABLE]

∎

Proof of Corollary 1.2.

The result holds trivially for $x>k$ by (1.6), so we may assume that $x\leq k$. Then, using Theorem 1.3 with $y=(\log k)^{3}$, together with Lemma 4.3 and our assumption on $x$ completes the proof. ∎

Acknowledgements

I would like to thank Emmanuel Kowalski for useful comments concerning the probabilistic random model for the Hecke eigenvalues in Section 3.