On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments
Jiseong Kim, Kunjakanan Nath
TL;DR
We study modular analogues of Hardy–Littlewood prime-tuple correlations for holomorphic cusp forms by examining correlations of Hecke eigenvalues at prime arguments weighted by the von Mangoldt function. The central object is $V_f(X;H)$, and we combine the circle method with Gallagher's lemma, zero-free regions for $L(s,f,χ)$, and moment hypotheses to bound these correlations; averaging over Dirichlet characters and forms strengthens the results. Under a Littlewood-type zero-free region, we obtain $V_f(X;H) \ll_A \frac{HX^2}{(\log X)^A}$ for $X^{2/3+ε} \le H \le X^{1-ε}$; with a fourth-moment hypothesis this extends to $X^{1/3+ε} \le H \le X^{1-ε}$. Averaging over the Hecke basis via the Petersson formula yields the same bound in the weight-aspect range $X^{ε} \le H \le X^{1-ε}$ for sufficiently large even $k$. These results provide modular analogues of averaged Hardy–Littlewood prime tuples in a cusp-form setting.
Abstract
Let $\{λ_f(n)\}_{n \geq 1}$ 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 $L(s, f, χ)$, where $χ$ is a Dirichlet character modulo $q$, that if $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ with $\varepsilon>0$, then for any $A\geq 1$, $$\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}}$$ 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 $A\geq 1$, $$ \sum_{f\in \mathcal{H}_k}ω_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}},$$ 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 $ω_f$ 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.