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On variants of Chebyshev's conjecture

Masatoshi Suzuki

TL;DR

The paper investigates sign patterns of weighted summatory functions of the von Mangoldt function and proves that constant-sign behavior, whether locally or on average, is intimately tied to the Riemann hypothesis and the generalized Riemann hypothesis for Dirichlet L-functions. By expressing these sums via integral transforms and explicit formulas, the author derives equivalences between RH/GRH and averaged sign conditions, and shows that the weighted sums are negative on average even without assuming RH/GRH. The results extend from the Riemann zeta function to general Dirichlet L-functions and further to averages over Dirichlet characters modulo q, yielding criteria for GRH in terms of sign constancy of arithmetic sums in progressions. The work highlights a novel connection between sign/bias phenomena in weighted prime sums and the location of zeros of L-functions, providing new avenues for RH/GRH criteria through average-sign analyses and harmonic-analytic representations.

Abstract

We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann hypothesis or the generalized Riemann hypothesis for Dirichlet $L$-functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average.

On variants of Chebyshev's conjecture

TL;DR

The paper investigates sign patterns of weighted summatory functions of the von Mangoldt function and proves that constant-sign behavior, whether locally or on average, is intimately tied to the Riemann hypothesis and the generalized Riemann hypothesis for Dirichlet L-functions. By expressing these sums via integral transforms and explicit formulas, the author derives equivalences between RH/GRH and averaged sign conditions, and shows that the weighted sums are negative on average even without assuming RH/GRH. The results extend from the Riemann zeta function to general Dirichlet L-functions and further to averages over Dirichlet characters modulo q, yielding criteria for GRH in terms of sign constancy of arithmetic sums in progressions. The work highlights a novel connection between sign/bias phenomena in weighted prime sums and the location of zeros of L-functions, providing new avenues for RH/GRH criteria through average-sign analyses and harmonic-analytic representations.

Abstract

We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann hypothesis or the generalized Riemann hypothesis for Dirichlet -functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average.

Paper Structure

This paper contains 20 sections, 11 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Results on the Riemann zeta function
  3. Results related to Chebyshev's conjecture
  4. Results on averages over Dirichlet characters
  5. Preliminaries
  6. Facts on prime numbers
  7. Riemann zeta function
  8. Dirichlet $L$-function
  9. Mellin transforms of nonnegative functions
  10. Proof of Theorems \ref{['thm_1']} and \ref{['thm_2']}
  11. Proof of Theorem \ref{['thm_1']}
  12. Proof of Corollary \ref{['cor_1']}
  13. Proof of Theorem \ref{['thm_2']}
  14. Proof of Theorems \ref{['thm_3']}, \ref{['thm_4']}, and \ref{['thm_5']}
  15. Proof of Theorem \ref{['thm_3']}
  16. ...and 5 more sections

Key Result

Theorem 1

The following three statements are equivalent:

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Proposition 1
  • Theorem 8
  • ...and 3 more