= Holomorphy of a Dirichlet series from bounded partial sums
{title2=$\sup_x|\sum_{n\le x}b_n|<\infty\implies\sum b_n n^{-s}\text{ holomorphic for }\operatorname{Re}s>0$}
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 $\operatorname{Re}s>0$, 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
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