Shifted Dirichlet series - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:14:28Z http://mathoverflow.net/feeds/question/6450 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6450/shifted-dirichlet-series Shifted Dirichlet series Dan Brumleve 2009-11-22T09:52:40Z 2009-11-22T11:15:52Z <p>If $\sum_{n=1}^\infty \frac{a_n}{n^s}$ converges, does $\sum_{n=1}^\infty \frac{a_n}{(n+1)^s}$ also converge?</p> http://mathoverflow.net/questions/6450/shifted-dirichlet-series/6454#6454 Answer by engelbrekt for Shifted Dirichlet series engelbrekt 2009-11-22T11:15:52Z 2009-11-22T11:15:52Z <p>Yes, because $(n+1)^{-s} = n^{-s} + sn^{-s-1} + O(|s|^2n^{-\sigma - 2})$. The first series necessarily converges in the open half-plane strictly to the right of s, and converges absolutely in the half-plane strictly to the right of s + 1. I hope I am not doing homework from a course in analytic number theory here.</p>