Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I\_s$ be defined as above.


Is $I\_s$ known to be convergent for any real number $s<1$ ?



On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
I claim that no such $s$ is known to exist. Indeed, define $\sigma\_c$ to be the abscissa of convergence of $I$. Then 


$$\sigma\_c = \limsup\_{x\rightarrow \infty} \frac{\log|\pi(x)-Li(x)|}{\log x}.$$ Since we do not know of any absolute $\theta<1$ such that $|\pi(x)-Li(x)|\ll x^{\theta}$, the claim follows.


