On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments

TL;DR

This paper investigates correlations of Fourier coefficients of holomorphic cusp forms at prime arguments, establishing bounds under certain hypotheses and averaging over forms, thus extending prime tuple conjecture analogues to modular forms.

Contribution

It provides new bounds for binary correlations of Fourier coefficients of cusp forms at primes, assuming zero-free regions and GRH-related hypotheses, and extends results to averaged forms.

Findings

Bounded sums of Fourier coefficient correlations under hypotheses

Extended results assuming GRH-related moment conditions

Averaged bounds over the space of cusp forms

Abstract

Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where $\chi$ is a Dirichlet character modulo $q$, that if $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ with $\varepsilon>0$, then for any $A\geq 1$, $$\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} \lambda_f(n)\Lambda(n)\lambda_f(m)\Lambda(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}}$$ holds. Moreover, under an additional hypothesis on the fourth moment of certain Dirichlet polynomials (which follows from GRH for $L(s, f)$), we show that the above result can be strengthened to hold in a wider range $X^{1/3+\varepsilon}\ll H \ll X^{1-\varepsilon}$. Finally, if we average over the forms $f$, then for $X^{\varepsilon}\ll H\ll X^{1-\varepsilon}$ and for any $A\geq 1$, $$ \sum_{f\in \mathcal{H}_k}\omega_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} \lambda_f(n)\Lambda(n)\lambda_f(m)\Lambda(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}},$$ where $\mathcal{H}_k$ is the Hecke basis for the space of holomorphic cusp forms of weight $k$ for the full modular group $\mathrm{SL}(2, \mathbb{Z})$ and $\omega_f$ are harmonic weights associated with $f\in \mathcal{H}_k$. These results may be viewed as modular analogues of the averaged forms of the Hardy--Littlewood prime tuple conjecture.