Sign changes in Mertens' first and second theorems

Abstract

We show that the functions $\sum_{p\leq x} (\log p)/p - \log x - E$ and $\sum_{p\leq x} 1/p - \log \log x -B$ change sign infinitely often, and that under certain assumptions, they exhibit a strong bias towards positive values. These results build on recent work of Diamond & Pintz and Lamzouri concerning oscillation of Mertens' product formula, and answers to the affirmative a question posed by Rosser and Schoenfeld.