Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On signs of coefficients of L-functions

Didier Lesesvre, Ming Ho Ng, Yingnan Wang

Abstract

We give a general lower bound on the frequency of sign changes in the real coefficients of L-functions of the Selberg class. We in particular recover existing results in the cases of GL(2) and GL(3), and obtain new bounds in the case of GSp(4).

On signs of coefficients of L-functions

Abstract

We give a general lower bound on the frequency of sign changes in the real coefficients of L-functions of the Selberg class. We in particular recover existing results in the cases of GL(2) and GL(3), and obtain new bounds in the case of GSp(4).
Paper Structure (14 sections, 10 theorems, 73 equations)

This paper contains 14 sections, 10 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. $L$-functions and the Selberg class
  3. Signs of coefficients
  4. Upper bound
  5. Preliminary lemmas
  6. Reducing to an integral via Perron formula
  7. Möbius to detect coprimality
  8. Moving the contours
  9. High parts on the right
  10. Horizontal parts
  11. Bulk on the left
  12. Lower bound
  13. Proof of the theorem
  14. An application to $\mathrm{GSp}(4)$

Key Result

Theorem 1

Consider an $L$-function with real coefficients $(A(m))_{m\geqslant 1}$ in the Selberg class as described above --- in particular a bound towards Lindelöf hypothesis with exponent $\theta$, a bound towards non-vanishing with exponent $\kappa$ and a Rankin-Selberg estimate. We assume that Then the number of sign changes in $(A(m))_{m\leqslant X}$ is at least for $\delta = \sqrt{2-2\kappa+\varepsi

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Proposition 1
  • Lemma 1: Subconvexity bound for partial sums
  • proof
  • Lemma 2: Removing congruences
  • ...and 9 more