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arXiv · 1410.3777

A bias in Mertens' product formula

Youness Lamzouri
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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.

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Youness Lamzouri. 2014-10-14. A bias in Mertens' product formula. https://arxiv.org/abs/1410.3777

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