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

Let $\{{\lambda }_f(n){\}}_{n\ge 1}$ 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 $X^{2\mathrm{/}3+\epsilon }\ll H\ll X^{1-\epsilon }$ with $\epsilon >0$, then for any $A\ge 1$, $$\sum _{1\le \mathrm{\mid }h\mathrm{\mid }\le H}\mid \sum _{\begin{array}{c}X<n,m\le 2X\\ n-m=h\end{array}}{\lambda }_f(n)\mathrm{\Lambda }(n){\lambda }_f(m)\mathrm{\Lambda }(m){\mid }^2{\ll }_A\frac{H{X}^2}{(\log X{)}^A}$$ 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 $X^{1\mathrm{/}3+\epsilon }\ll H\ll X^{1-\epsilon }$. Finally, if we average over the forms $f$, then for $X^{\epsilon }\ll H\ll X^{1-\epsilon }$ and for any $A\ge 1$, $$\sum _{f\in {\mathcal{H}}_k}{\omega }_f\sum _{1\le \mathrm{\mid }h\mathrm{\mid }\le H}\mid \sum _{\begin{array}{c}X<n,m\le 2X\\ n-m=h\end{array}}{\lambda }_f(n)\mathrm{\Lambda }(n){\lambda }_f(m)\mathrm{\Lambda }(m){\mid }^2{\ll }_A\frac{H{X}^2}{(\log X{)}^A},$$ where ${\mathcal{H}}_k$ is the Hecke basis for the space of holomorphic cusp forms of weight $k$ for the full modular group $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ 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.