A bias in Mertens’ product formula

Abstract

Rosser and Schoenfeld remarked that the product p≤x (1 − 1/p) −1 exceeds e γ log x for all 2 ≤ x ≤ 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 p≤x (1 − 1/p) −1 and e γ log x which explains the computations of Rosser and Schoenfeld.

The Erdős conjecture for primitive sets

“…The proof depends strongly on a recent result of Lamzouri [13], who was interested in the "Mertens race" between p≤x (1 − 1/p) and 1/(e γ log x).…”

“…To complete the proof, we use a result of Lamzouri [13] relating the Mertens inequality to the race between π(x) and li(x), studied by Rubinstein and Sarnak [18]. Under the assumption of RH and LI, he proved that the set N of real numbers x satisfying We note that if a prime p = p n ∈ N , then for p = p n+1 we have [p, p ) ⊂ N because the prime product on the left-hand side is constant on…”

Primes in prime number races

“…The "Mertens race" between e γ log x and this product of Mertens is mathematically analagous to the race between li(x) and π(x). Recent analysis of Lamzouri [10] implies, conditionally on RH and LI, that the normalized error function…”

“…is nonnegative. To show the density of N exists we follow the general plan laid out by Lamzouri [10], who proved analogous results for the Mertens race, with some important modifications. 5.1.…”

“…The fact that this upper bound tends to 0 easily implies condition (a) of Theorem 2.1. Finally, condition (b) is satisfied by a similar argument as in the proof of Theorem 2.2, using work of Lamzouri[10] on the limiting logarithmic distribution of E M (x).…”