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A Brun-Titchmarsh inequality for weighted sums over prime numbers

Jan Büthe — 2014

Acta Arithmetica

We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.

On the first sign change in Mertens' theorem

Jan Büthe — 2015

Acta Arithmetica

The function $\sum_{p \leq x} 1 / p - l o g l o g (x) - M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

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