On variants of Chebyshev's conjecture

TL;DR

This paper links the sign behavior of specific summatory functions of the von Mangoldt function to the Riemann hypothesis and its generalization, showing average sign constancy despite individual challenges.

Contribution

It establishes that average sign constancy of certain weighted summatory functions implies the Riemann hypothesis or its generalization.

Findings

Sign constancy implies Riemann hypothesis

Summatory functions have constant signs on average

Provides a new approach to understanding RH

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.