# Positivity of an integral

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).

• 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? – Greg Martin Mar 31 '19 at 18:26
• Context: $J(x)$ appears in old papers of Schoenfeld and Nicolas and it is related to the sum of prime reciprocals – EGME Mar 31 '19 at 18:31
• Instead of clarifying in comments, please edit the original post to include the improvements. – Greg Martin Mar 31 '19 at 18:32