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This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ converges for some real $s=\sigma<1$. Does it then follow that $I_s$ converges for $s=\sigma+it$ for any real $t$ ?

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  • $\begingroup$ In fact I believe that this question, or more precisely Greg Martin's answer to it, logically precedes the quoted previous question. See my response to the previous question: it contains more detail than the accepted answer (to that question). $\endgroup$
    – GH from MO
    Commented Aug 9, 2019 at 4:13

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Not quite: if a Dirichlet integral of this shape converges at $\sigma+it$, then it converges in the half-plane $\Re s > \sigma$, but not necessarily on the line $\Re s=\sigma$.

The proof for Dirichlet series in place of Dirichlet integrals is standard (see for example Theorem 1.1 of Montgomery/Vaughan. The proof should adapt easily to Dirichlet integrals.

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