A bias in Mertens' product formula

TL;DR

This paper investigates the bias in Mertens' product formula, revealing a strong tendency for the product to exceed the logarithmic benchmark, which explains previous computational observations.

Contribution

It demonstrates, under certain hypotheses, a significant bias in the comparison between Mertens' product and the logarithmic function, extending understanding of their relationship.

Findings

The product tends to exceed $e^{b3}\, ext{log}\,x$ more often than not.

There is a strong bias in the race between the product and the logarithmic benchmark.

The bias explains previous computational results by Rosser and Schoenfeld.

Abstract

Rosser and Schoenfeld remarked that the product $\prod_{p\leq x}(1-1/p)^{-1}$ exceeds $e^{\gamma} \log x$ for all $2\leq x\leq 10^8$, and raised the question whether the difference changes sign infinitely often. This was confirmed in a recent paper of Diamond and Pintz. In this paper, we show (under certain hypotheses) that there is a strong bias in the race between the product $\prod_{p\leq x}(1-1/p)^{-1}$ and $e^{\gamma}\log x$ which explains the computations of Rosser and Schoenfeld.