On the mean values of the error terms in Mertens' theorems

Tianyu Zhao

doi:10.1007/s40993-025-00640-y

paper

On the mean values of the error terms in Mertens' theorems

arXiv:2411.18903 · doi:10.1007/s40993-025-00640-y

Abstract

For $i \in {1, 2, 3}$ , let $E_{i} (x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i \in {1, 2}$ the Riemann hypothesis is equivalent to the condition $\int_{2}^{X} E_{i} (x) d x > 0$ for all $X > 2$ , and we examine assumptions under which the equivalence also holds for $i = 3$ . In addition, we extend our results to analogues of Mertens' theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.

23 pages; accepted version