Let $f(x)=x\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ for all $x\geq 2$? Note that if $f$ were continuous, this would be the case. $\theta$ is the Chebyshev theta. The integral appears in Rosser and Schoenfeld, and also in a paper by Nicolas, and it is related to the sum of prime reciprocals. Its positivity is equivalent to the Riemann Hypothesis (reference NA).

1$\begingroup$ As with your previous question, this post is greatly lacking in context (including the origin of that particular integral). Furthermore, it's probably unlikely that your "suppose" assertion is actually true; do you want the answer to your question to be "yes", as would be vacuously valid? $\endgroup$ – Greg Martin Mar 31 '19 at 18:26

$\begingroup$ Context: $J(x)$ appears in old papers of Schoenfeld and Nicolas and it is related to the sum of prime reciprocals $\endgroup$ – EGME Mar 31 '19 at 18:31

4$\begingroup$ Instead of clarifying in comments, please edit the original post to include the improvements. $\endgroup$ – Greg Martin Mar 31 '19 at 18:32