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Where can I find the theory of abcissa of convergence for integrals necessary to understand ChenClass answer to

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

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Note that the integral I is not a Dirichlet integral since Li is not a summatory function. In other words sI is not a Dirichlet series.

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  • $\begingroup$ this is a duplicate of mathoverflow.net/q/342835/11260 , please don't post the same question twice. $\endgroup$
    – Carlo Beenakker
    Commented Oct 1, 2019 at 11:41
  • $\begingroup$ I did not know the other question. My formulation is clearer I think, $\endgroup$
    – Patrick Sole
    Commented Oct 1, 2019 at 11:46
  • $\begingroup$ It works exactly the same way as for Dirichlet series, indeed $Li(x) = \sum_{2 \le n \le x} \frac1{\log n} + O(1)$ and $s \int_1^\infty (\pi(x)- \sum_{2 \le n \le x} \frac1{\log n})x^{-s-1} dx$ is a Dirichlet series. Of course once the basics are clear what we care is how it is related to and affected by the prime number theorem and the Riemann hypothesis, the abscissa of convergence being the main topic of every book on $\zeta(s)$ $\endgroup$
    – reuns
    Commented Oct 1, 2019 at 11:55
  • $\begingroup$ Would you have a book, an article to refer me to? Note that I cannot open the link underline in blue, being in China at the moment. $\endgroup$
    – Patrick Sole
    Commented Oct 1, 2019 at 12:10
  • $\begingroup$ They are on libgen and Titchmarsh, Apostol, Montgomery and all others are on google $\endgroup$
    – reuns
    Commented Oct 1, 2019 at 12:12

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