For the first two Mertens error terms, the Riemann hypothesis is equivalent to the positivity of their integral from 2 to X for every X>2.
Oscillation of Mertens’ product formula
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On the mean values of the error terms in Mertens' theorems
For the first two Mertens error terms, the Riemann hypothesis is equivalent to the positivity of their integral from 2 to X for every X>2.