On the exponent of distribution for convolutions of $\operatorname{GL}(2)$ coefficients to smooth moduli

Rongjie Yin; Tengyou Zhu

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arXiv · 2605.09322

On the exponent of distribution for convolutions of $\operatorname{GL}(2)$ coefficients to smooth moduli

Rongjie Yin·Tengyou Zhu

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Abstract

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{70}$ when the modulus $q$ is square-free and has only sufficiently small prime factors.

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Rongjie Yin, Tengyou Zhu. 2026-05-10. On the exponent of distribution for convolutions of $\operatorname{GL}(2)$ coefficients to smooth moduli. https://arxiv.org/abs/2605.09322

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