About the sum of prime reciprocals

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 $$\log\log x$$.

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