Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chebyshev's bias for modular forms

Shin-ya Koyama, Arshay Sheth

TL;DR

This paper investigates Chebyshev-type biases in the signs of Fourier coefficients of modular forms. It demonstrates that, under the Deep Riemann Hypothesis framework, the bias is completely governed by $m(f)=\operatorname{ord}_{s=1/2}L(s,f)$, yielding a positive bias when $m(f)=0$ and a negative bias when $m(f)\ge 1$, with refinements under GRH. The authors provide explicit computations of $m(f)$ in several level-1 examples, and verify super-positivity and Mizumoto positivity for Ramanujan-type forms, thereby giving concrete, unconditional (in some cases) manifestations of the bias phenomenon. The work connects central-value vanishing to Chebyshev-type bias through Satake parameter analysis and automorphic $L$-functions, offering a precise, testable framework for bias in modular forms and potentially broader automorphic contexts.

Abstract

We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $Γ_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples.

Chebyshev's bias for modular forms

TL;DR

This paper investigates Chebyshev-type biases in the signs of Fourier coefficients of modular forms. It demonstrates that, under the Deep Riemann Hypothesis framework, the bias is completely governed by , yielding a positive bias when and a negative bias when , with refinements under GRH. The authors provide explicit computations of in several level-1 examples, and verify super-positivity and Mizumoto positivity for Ramanujan-type forms, thereby giving concrete, unconditional (in some cases) manifestations of the bias phenomenon. The work connects central-value vanishing to Chebyshev-type bias through Satake parameter analysis and automorphic -functions, offering a precise, testable framework for bias in modular forms and potentially broader automorphic contexts.

Abstract

We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on . Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the -function at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples.

Paper Structure

This paper contains 8 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result.
  3. Proof of Theorem \ref{['mainthm6']}
  4. Explicit examples
  5. Vanishing caused by root number considerations
  6. Super-positivity
  7. Mizumoto's positivity
  8. Modular forms with real Fourier coefficients

Key Result

Theorem 1.5

Let $f \in S_k(\Gamma_0(N))$ be a cuspidal newform. Assume Conjecture DRH for $L(s, f)$ and let $m(f)=\mathop{\mathrm{ord}}_{s=1/2} L(s, f)$. Then there exists a constant $c_f$ such that In particular, in the sense of Definition Chebyshev, we conclude that

Theorems & Definitions (27)

  • Conjecture 1.1: Kaneko--Koyama--Kurokawa KanekoKoyamaKurokawa2022
  • Definition 1.2: Aoki--Koyama AokiKoyama2023
  • Example 1.3: Aoki--Koyama AokiKoyama2023
  • Example 1.4: Koyama--Kurokawa KK22
  • Theorem 1.5
  • Theorem 2.1: Chebyshev's bias for Satake parameters Sheth24B
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 17 more