Very particular limsup of an integral

Question

Maybe its a very hard problem, but does someone know if exists a positive real function $g(x)$ such that: $\displaystyle \limsup_{x\rightarrow +\infty}\frac{\displaystyle \int_{x}^{+\infty} \frac{W(t)}{t^{2}}dt}{g(x)} >1$, where $W(t)=\pi(t)-li(t)$, ?

Answer 1

The general approach would be the following: First express $W$ using roots of $\zeta$. Then show that everything converges so well that you can interchange the sum over zeros with the integral over $t$ to get an explicit formula for $\int_x^\infty\frac{W(t)}{t^2}\;dt$. Finally use Turans power sum method to show that this integral becomes large and positive infinitely often. This strategy is well explained in Turan's book "A new method in analysis with applications".

The details become messy because the explicit formula will involve expressions of the form $\mathrm{li}(x^\rho)$, which you have to approximate before you apply general theorems. As an alternative you could first consider the integral $\int_x^\infty\frac{\tilde{W}(t)}{t^2}\;dt$, where $\tilde{W}=\Psi(x)-x$. In this case the explicit formula contains only terms of the form $\frac{x^\rho}{\rho}$, which are much easier to handle. Then you need some approximation to show that the existence of large positive values of the integral for $\tilde{W}$ implies the existence of large positive values of the integral for $W$.