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

TL;DR

This paper investigates the relationship between the Riemann hypothesis and the integral behavior of error terms in Mertens' theorems, extending results to prime sums with characters and progressions.

Contribution

It establishes the equivalence of the Riemann hypothesis with positivity conditions on error term integrals for two of Mertens' theorems and explores extensions to related prime sum contexts.

Findings

Riemann hypothesis equivalent to integral positivity of error terms for two Mertens' theorems

Conditions identified under which the equivalence extends to the third theorem

Results extended to prime sums twisted by characters and in arithmetic progressions

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 $\int_2^X E_i(x) \:\mathrm{d}x>0$ 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.