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 $\theta(x)-x=0$? 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 $\log\log x$.

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 $\log\log x$.

Let $B$ be the Meissel-Mertens constant., $\theta$ the Chebyshev theta. Let $S(x)=\sum_{p\le x}1/p$. 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 $\theta(x)-x=0$? 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 $\log\log x$.

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 $\theta(x)-x=0$? 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 $\log\log x$.