Absolute convergence at one point forces absolute and locally uniform convergence on closed half-planes to the right

Write ${\sigma }_0:=\mathrm{\Re }{s}_0$. Since $\sum \mathrm{\mid }{a}_n\mathrm{\mid }{n}^{-{\sigma }_0}$ converges, for every $s$ with $\mathrm{\Re }s\ge {\sigma }_0+\epsilon$ one has $\mathrm{\mid }{a}_n{n}^{-s}\mathrm{\mid }\le \mathrm{\mid }{a}_n\mathrm{\mid }{n}^{-{\sigma }_0-\epsilon }\le \mathrm{\mid }{a}_n\mathrm{\mid }{n}^{-{\sigma }_0}\mathrm{.}$ Thus [L1] gives absolute and locally uniform convergence of the original series on the stated closed half-plane.

For large $n$ one has $\log n\le n^{\epsilon \mathrm{/}2}$, so on the same region $\mathrm{\mid }{a}_n(\log n)n^{-s}\mathrm{\mid }\le \mathrm{\mid }{a}_n\mathrm{\mid }{n}^{-{\sigma }_0-\epsilon \mathrm{/}2}\mathrm{.}$ The majorant series on the right converges because it is termwise bounded by $\sum \mathrm{\mid }{a}_n\mathrm{\mid }{n}^{-{\sigma }_0}$. Therefore [L1] also gives local uniform convergence of the derivative series.

The partial sums are holomorphic, the derivative series converges locally uniformly, and the original series converges at every point of the half-plane from step 1.1. Hence [L2] yields termwise differentiation there.