Holomorphy of a Dirichlet series from bounded partial sums (${\sup }_x|{\sum }_{n\le x}{b}_n|<\infty ⟹\sum b_n{n}^{-s}\text{ holomorphic for }\mathrm{Re}s>0$)

Writing $B(x)={\sum }_{n\le x}{b}_n$, the Abel summation formula gives $\sum b_n{n}^{-s}=s{\int }_1^{\infty }B(x)x^{-s-1}dx$. If $B$ is bounded, the integral and its differentiated integrands converge uniformly on compact subsets of $\mathrm{Re}s>0$, giving a holomorphic function. A nonprincipal Dirichlet character has bounded partial sums because it is periodic and its sum over a period is zero.

// Target: algebra.bigb