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On the average least negative Hecke eigenvalue

Jackie Voros

TL;DR

This work analyzes the average behavior of the first sign change in classical Hecke eigenvalues for weight $k$ newforms, distinguishing the first negative prime eigenvalue $p_f$ from the first negative eigenvalue $n_f$. It proves that the average $p_f$ converges to the Erdős-type constant $\sum_{i=1}^{\infty} p_i/2^i$ under GRH for symmetric power $L$-functions, while the average $n_f$ converges unconditionally to a finite, explicit constant expressed via local $p$-adic Plancherel measures; both results are uniform in the weight and level through a robust large-sieve framework. The proofs blend Serre-type equidistribution for Hecke angles, the Sato–Tate distribution at primes not dividing the level, and Atkin–Lehner considerations at primes dividing the level with square-free $N$, together with a weighted Linnik-type large sieve and amplifier techniques to bound large sign-change positions. The results illuminate the sign distribution of Hecke eigenvalues and create a precise analogue of the least quadratic non-residue problem in the automorphic setting, highlighting the power of uniform sieve methods in analytic number theory.

Abstract

We show that the first sign change of Hecke eigenvalues of classical newforms has a finite mean, which we also compute. We distinguish between the first negative prime Hecke eigenvalue, and the first negative Hecke eigenvalue. This problem can be considered to be an analogue of the least quadratic non-residue problem, of which the average was explored by Erdős in 1961. In fact, the average least negative prime Hecke eigenvalue has the same value as the average least quadratic non-residue, under GRH. To compute these averages, we develop large sieve inequalities that are uniform in both the weight and level aspect.

On the average least negative Hecke eigenvalue

TL;DR

This work analyzes the average behavior of the first sign change in classical Hecke eigenvalues for weight newforms, distinguishing the first negative prime eigenvalue from the first negative eigenvalue . It proves that the average converges to the Erdős-type constant under GRH for symmetric power -functions, while the average converges unconditionally to a finite, explicit constant expressed via local -adic Plancherel measures; both results are uniform in the weight and level through a robust large-sieve framework. The proofs blend Serre-type equidistribution for Hecke angles, the Sato–Tate distribution at primes not dividing the level, and Atkin–Lehner considerations at primes dividing the level with square-free , together with a weighted Linnik-type large sieve and amplifier techniques to bound large sign-change positions. The results illuminate the sign distribution of Hecke eigenvalues and create a precise analogue of the least quadratic non-residue problem in the automorphic setting, highlighting the power of uniform sieve methods in analytic number theory.

Abstract

We show that the first sign change of Hecke eigenvalues of classical newforms has a finite mean, which we also compute. We distinguish between the first negative prime Hecke eigenvalue, and the first negative Hecke eigenvalue. This problem can be considered to be an analogue of the least quadratic non-residue problem, of which the average was explored by Erdős in 1961. In fact, the average least negative prime Hecke eigenvalue has the same value as the average least quadratic non-residue, under GRH. To compute these averages, we develop large sieve inequalities that are uniform in both the weight and level aspect.

Paper Structure

This paper contains 13 sections, 13 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results
  2. Motivation
  3. Proof of main results
  4. Proof of Theorem \ref{['thrm:averagepf']}
  5. Proof of Theorem \ref{['thrm:averagenf']}
  6. Proof of auxiliary results
  7. Proof of Lemma \ref{['lem:pf half']}
  8. Proof of Corollaries \ref{['cor: large sieve']} and \ref{['cor: nf large sieve']}
  9. Proof of weighted large sieve and statistical results
  10. Proof of Theorem \ref{['thrm: linnik nf']}
  11. Proof of Theorem \ref{['thrm:weighted linnik sieve']}
  12. Proof of Corollary \ref{['cor:weighted polynomial sieve']}
  13. A complete sieve

Key Result

Theorem 1.1

Let $p_i$ denote the $i^{th}$ prime, and let $p_f$ be the least prime such that $\lambda_f(p_f) < 0$. Assuming GRH we have, as $k+N$ tends to infinity over even $k$ and square-free $N$.

Figures (2)

  • Figure 1: Distribution of $\lambda_f(p)$ as $k,N \to \infty$ (coloured lines), in comparison to the distribution of $\lambda_f(p)$ as $p \to \infty$ (black line).
  • Figure 2: $F = - \frac{3}{4}X_0 + \frac{1}{2}X_1 + \frac{1}{4}X_2$, $\widetilde{F}=-\frac{3}{4}X_0 + \frac{1}{2}X_1 - \frac{1}{4}X_2$, compared with $\textrm{sgn}(2 \cos \theta)$ in red.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • Corollary 3.2
  • Theorem 3.3: Ser
  • Corollary 3.4
  • Proposition 4.1: Ser
  • ...and 6 more