 Solution

Use $q=e^{2\pi i\tau }$, so $g(iy)={\sum }_{n\ge 1}{b}_n{e}^{-2\pi ny}$. Put $u=\mathrm{Re}s$. For $u>\max \{0,\sigma +1\}$,The Gamma integral and absolute convergence therefore justify exchanging summation and integration. For every positive integer $n$, substitution $t=2\pi ny$ givesSumming these identities proves the Mellin transform formula for the Dirichlet series:The power of the positive real variable $y$ uses the real logarithm; the initial half-plane stated above is a sufficient one even when $\sigma$ is negative.