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 $$\Re \left({z}\right) \geq 1 \ $$.