A bias in Mertens' product formula

Abstract

Rosser and Schoenfeld remarked that the product ${\prod }_{p\le x}(1-1/p{)}^{-1}$ exceeds $e^{\gamma }\log x$ for all $2\le x\le 10^8$, 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 ${\prod }_{p\le x}(1-1/p{)}^{-1}$ and $e^{\gamma }\log x$ which explains the computations of Rosser and Schoenfeld.

arXiv comment13 pages. Fixed a mistake in Lemma 2.4. To appear in IJNT

References in corpus (2)

• Comparative prime number theory: A survey
• Limiting distributions of the classical error terms of prime number theory