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Let $B$ be the Meissel-Mertens constant., $\theta$ the Chebyshev theta. Let $S(x)=\sum_{p\le x}1/p$, $p$ prime. Let $M(x)=S(x)-\log\log x-B$. Robin, and later Diamond and Pintz, showed that $M(x)$ changes sign infinitely often. Is $M(x)>0$ when $\theta(x)-x=0$, more specifically, in the case when $\theta(x)-x$ is generally growing? The intuition for this is that $\theta$ grows relative to $x$ when there are lots of primes close to each other, which is also when $S(x)$ grows relative to $\log\log x$.

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    $\begingroup$ I would bet that the signs of $M(x)$ and $\theta(x)-x$ are pretty independent of each other. $\endgroup$ – GH from MO Mar 31 '19 at 20:54
  • $\begingroup$ Have you tried to compute the integral $\int_{c}^{t}M(x)(\theta(x)-x)dx$ for rather large values of $t$? $\endgroup$ – Sylvain JULIEN Mar 31 '19 at 21:13

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