Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms

TL;DR

This paper investigates the distribution and sign changes of Fourier coefficients of $ ext{ell}$-fold product $L$-functions at integers represented by quadratic forms, providing explicit bounds and generalizations.

Contribution

It establishes explicit bounds for the Fourier coefficients' sum over integers represented by quadratic forms and generalizes sign change results to multiple forms, improving with class number.

Findings

Derived explicit upper bounds for the sum of coefficients.

Proved bounds for the first sign change of the coefficients.

Generalized sign change results across multiple quadratic forms.

Abstract

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases.