The paper proves equivalences between the Riemann hypothesis and persistent inequalities of normalized error terms in weighted prime counting functions, and computes conditional logarithmic densities of sign agreements such as ≈0.9865 for specific pairs.
Sign changes in Mertens' first and second theorems
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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.
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Correlations of error terms for weighted prime counting functions
The paper proves equivalences between the Riemann hypothesis and persistent inequalities of normalized error terms in weighted prime counting functions, and computes conditional logarithmic densities of sign agreements such as ≈0.9865 for specific pairs.