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

TL;DR

This work analyzes the Fourier coefficients of odd $\ell$-fold product $L$-functions at integers represented by binary quadratic forms. By expressing the relevant summatory function over the sparse set of represented integers and employing a smoothing method together with a detailed Euler-product decomposition into symmetric-power $L$-functions, the authors obtain an explicit bound for the sum in terms of the weight $k$, level $N$, discriminant $D$, and the form's class structure. They also derive a quantitative first sign-change bound for the coefficients, with refinements when considering all reduced forms of a fixed discriminant and nontrivial class numbers. The results advance the understanding of higher-rank automorphic $L$-functions over arithmetic sets defined by quadratic forms and provide tools for analyzing sign fluctuations and distribution on sparse quadratic-representation sets.

Abstract

Let $f \in S_{k}(Γ_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $λ_{f \otimes \cdots \otimes_{\ell} f}(n)$ 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.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Remark 1.3
• Remark 2.1
• Lemma 2.1: Lalit-M
• Lemma 2.2
• proof
• Lemma 2.3