Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)>J(2)>0$, and let $J(x)>0$ for $x>2$. Does it follow that $K(x)>0$ for $x>2$? If this is not the case, could someone supply such an $S$ for which this fails?

Context: Integrals like this are closely related to integrals that appear in the study of sum of prime reciprocals in Rosser and Schoenfeld’s well known paper on formulas for some functions of prime numbers. The positivity of an integral like $J$ is equivalent to the Riemann Hypothesis, and the positivity of an integral like $K$ is conjectured to be equivalent.

like$J$ is equivalent to the Riemann Hypothesis"? Can we get a precise statement? Citation? $\endgroup$ – Stopple Apr 5 '19 at 23:12