Sign changes in Mertens' first and second theorems

Abstract

We show that the functions ${\sum }_{p\le x}(\log p)/p-\log x-E$ and ${\sum }_{p\le x}1/p-\log \log x-B$ 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.

13 pages

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