On the first sign change in Mertens' Theorem

TL;DR

This paper proves that the first sign change in the Mertens' theorem-related function occurs much earlier than previously known, providing a significant advancement in understanding its behavior.

Contribution

The paper establishes an explicit upper bound for the first sign change in the Mertens' function, improving upon prior results by showing it occurs before a specific exponential threshold.

Findings

First sign change occurs before exp(495.7028)

Provides explicit bounds for sign change timing

Enhances understanding of Mertens' theorem behavior

Abstract

The function $\sum_{p\leq x} \frac{1}{p} - \log\log(x) - M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before $\exp(495.702833165)$.