Problem 1732 (difficulty: 9/10)

Assume that the Dirichlet series \(\displaystyle \displaystyle f(s)=\sum_{n=1}^\infty\frac{a_n}{n^s}\) absolutely converges for \(\displaystyle \re s\ge1\) and let \(\displaystyle X>0\) be real. Find the following integrals:

\(\displaystyle \lim_{h\to\infty}\frac1{2\pi i}\int_{\re s=1, |\im s|\le h}f(z)\frac{X^s}{s}\ds \qquad \frac1{2\pi i}\int_{\re s=1}f(z)\frac{X^s}{s^2}\ds \qquad \frac1{2\pi i}\int_{\re s=1}f(z)\frac{X^s}{s(s+1)}\ds \)

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