While self studying analytic proof of Prime Number Theorem from Apostol Introduction to analytic number theory , I couldn't think about a deduction in theorem contour integral representation of $\psi_1(x) $ / ($x^2$) .

My only doubt in this theorem is how Apostol writes $\sum_{n=1}^{\infty} \Lambda(n) / n^c $ to be absolutely convergent if $c>1$.

Can someone please tell how this is true?

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    $\begingroup$ $|\Lambda(n)|\le\log n$, right? and $\sum (\log n)/n^c$ is absolutely convergent for $c>1$, right? $\endgroup$ – Gerry Myerson Jan 13 '20 at 2:16

Notice that the Dirichlet series in question is equal to $-\zeta'(c)/\zeta(c)$, i.e. the negative of the logarithmic derivative of $\zeta(c)$. As $\zeta(s)$ converges absolutely for $\operatorname{Re}(s)>1$, the series does too.


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