A bias in Mertens' product formula

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.