Sign changes in Mertens' first and second theorems

TL;DR

This paper proves that certain functions related to Mertens' theorems change sign infinitely often and tend to be positive under some assumptions, extending recent oscillation results and answering longstanding questions.

Contribution

It establishes the infinite sign-changing nature of specific Mertens-related functions and demonstrates their bias towards positive values under certain hypotheses, advancing understanding of prime number distributions.

Findings

Functions change sign infinitely often

Functions tend to be positive under assumptions

Builds on recent oscillation results

Abstract

We show that the functions $\sum_{p\leq x} (\log p)/p - \log x - E$ and $\sum_{p\leq x} 1/p - \log \log x -B$ change sign infinitely often, and that under certain assumptions, they exhibit a strong bias towards positive values. These results build on recent work of Diamond & Pintz and Lamzouri concerning oscillation of Mertens' product formula, and answers to the affirmative a question posed by Rosser and Schoenfeld.