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First moment of Hecke eigenvalues at the integers represented by binary quadratic forms

Manish Kumar Pandey, Lalit vaishya

Abstract

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum; \begin{equation*} \begin{split} S(f, \mathcal{Q}; X ) &:= \sideset{}{^{\flat }}\sum_{n= \mathcal{Q}(\underline{x}) \le X \atop \gcd(n,N) =1 } λ_{f}(n), \end{split}\end{equation*} where $\flat$ means that sum runs over the square-free positive integers, $λ_{f}(n)$ denotes the normalised $n^{\rm th}$ Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup $Γ_{0}(N)$ and $\mathcal{Q}$ is a primitive integral positive-definite binary quadratic forms of fixed discriminant $D<0$ with the class number $h(D)=1$. As a consequence, we determine the size, in terms of conductor of associated $L$-function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by $\mathcal{Q}$. This work is an improvement and generalisation of the previous results.

First moment of Hecke eigenvalues at the integers represented by binary quadratic forms

Abstract

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum; \begin{equation*} \begin{split} S(f, \mathcal{Q}; X ) &:= \sideset{}{^{\flat }}\sum_{n= \mathcal{Q}(\underline{x}) \le X \atop \gcd(n,N) =1 } λ_{f}(n), \end{split}\end{equation*} where means that sum runs over the square-free positive integers, denotes the normalised Fourier coefficients of a Hecke eigenform of integral weight for the congruence subgroup and is a primitive integral positive-definite binary quadratic forms of fixed discriminant with the class number . As a consequence, we determine the size, in terms of conductor of associated -function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by . This work is an improvement and generalisation of the previous results.
Paper Structure (4 sections, 6 theorems, 79 equations)

This paper contains 4 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction and statements of the results
  2. Key Ingredients
  3. Mean value of a multiplicative function over the integers represented by a reduced form of discriminant $D$ with $h(D) =1$
  4. Proof of Results

Key Result

Theorem 1.1

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalised Hecke eigenform and $\mathcal{Q}$ be a reduced form of discriminant $D$ with the class number $h(D)=1$. For sufficiently large $X>0$ and any arbitrary small $\epsilon>0,$ we have where the implied constant is absolute.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 1 more