Sign changes in the prime number theorem

TL;DR

This paper establishes a new lower bound on the frequency of sign changes in the difference between the Chebyshev function and the identity, linking it to the zeros of the Riemann zeta-function, thus advancing understanding of prime distribution fluctuations.

Contribution

It improves the lower bound on the asymptotic density of sign changes in $ heta(x) - x$, connecting it to the lowest non-trivial zero of the Riemann zeta-function, refining previous results.

Findings

Lower bound on sign changes growth rate established

Connection between sign changes and zeros of zeta-function demonstrated

Enhancement over previous bounds by Kaczorowski

Abstract

Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.