# Regarding convergence of a series related to Mangoldt $\Lambda$ function used in PNT.

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?

• $|\Lambda(n)|\le\log n$, right? and $\sum (\log n)/n^c$ is absolutely convergent for $c>1$, right? – 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.