Ingham's Theorem on Convergent Dirichlet Series

Theorem
Suppose $$|a_n| \leq 1 \ $$, and form the series $$\sum_{n=1}^\infty a_n n^{-z} \ $$ which converges to an analytic function $$F(z) \ $$ for $$\Re \left({z}\right) > 1 \ $$, where $$\Re \left({z}\right)$$ is the real part of $$z \ $$.

If $$F(z) \ $$ in analytic throughout $$\Re \left({z}\right) \geq 1 \ $$, then $$\sum_{n=1}^\infty a_n n^{-z} \ $$ converges throughout $$\Re \left({z}\right) \geq 1 \ $$.

Proof
Fix a $$w \ $$ in $$\Re \left({w}\right) \geq 1 \ $$. Then $$F(z+w) \ $$ is analytic in $$\Re \left({z}\right) \geq 0 \ $$. Choose an $$R \geq 1 \ $$ and determine $$\delta = \delta (R) > 0, \delta \leq \tfrac{1}{2} \ $$ and an $$M = M(R) \ $$ so that $$F(z+w) \ $$ is analytic and bounded by $$M \ $$ in $$-\delta \leq \Re \left({z}\right), |z| \leq R \ $$.

Now form the counterclockwise contour $$\Gamma \ $$ as the arc $$|z|=R, \Re \left({z}\right) > - \delta \ $$ and the segment $$\Re \left({z}\right) = -\delta, |z| \leq R \ $$. We denote by $$A, B \ $$ respectively, the parts of $$\Gamma \ $$ in the right and left half-planes.

By the residue theorem,

$$2\pi i F(w) = \oint_{\Gamma} F(z+w) N^z \left({ \frac{1}{z} + \frac{z}{R^2} }\right) dz \ $$