# On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

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 $$s<1$$ ?

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.
The following answer is essentially the same as the one given by ChenClass, but it provides more detail. I claim that currently it is unknown whether $$I_s$$ converges for some real number $$s_0<1$$.
Assume that $$I_s$$ converges for some real number $$s_0<1$$. Adapting the proof of Theorem 1.1 in "Montgomery-Vaughan: Multiplicative number theory I", we see that $$I_s$$ converges locally uniformly in the half-plane $$H=\{s:\Re(s)>s_0\}$$. In particular, by Morera's theorem, $$I_s$$ is analytic in $$H$$. Using the explicit formulae on Page 465 of the same book, we infer that $$\log(\zeta(s)(s-1))$$ has an analytic continuation to $$H$$. Equivalently, $$\zeta(s)$$ has no zero on $$H$$. This is currently unknown, so at the moment we cannot prove that $$I_s$$ converges for some real number $$s_0<1$$.
On the other hand, the Riemann Hypothesis implies that $$I_s$$ converges in the half-plane $$\{s:\Re(s)>1/2\}$$, so at the moment we cannot disprove either that $$I_s$$ converges for some real number $$s_0<1$$.