A bias in Mertens' product formula

2014/10/14 by Youness Lamzouri, Lamzouri, Youness · 1 citation
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1410.3777

openalex publication_date 2014/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Rosser and Schoenfeld remarked that the product ∏_{p≤ x}(1-1/p)^-1 exceeds e^γ log x for all 2≤ x≤ 10⁸, 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 ∏_{p≤ x}(1-1/p)^-1 and e^γlog x which explains the computations of Rosser and Schoenfeld.

Citations

• Approximate formulas for some functions of prime numbers
• Limiting distributions of the classical error terms of prime number theory

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