On the exponent of distribution for convolutions of $\mathrm{GL(2)}$ coefficients to smooth moduli
Rongjie Yin
TL;DR
The paper studies the exponent of distribution for the GL(2) convolution $f(n)=(\lambda_f*1)(n)$ in arithmetic progressions with square-free moduli. It advances prior prime-modulus results by developing sharp bounds for bilinear sums with GL(2) coefficients, employing the $q$-van der Corput method together with Poisson and Voronoi summation to control oscillations. The main result shows that for square-free $q$ up to $X^{12/23-\varepsilon}$, the expected asymptotic holds with error $X/q(\log X)^{-A}$, giving the exponent $\theta_3 \ge 1/2+1/46$ for this GL(2) convolution. This extends distribution results to smooth moduli and demonstrates how bilinear form bounds for Kl_3 sums can unlock stronger progressions results in the GL(2) setting.
Abstract
Let $(λ_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $λ_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{46}$ when the modulus $q$ is square-free.