Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series

Sondow, Jonathan

doi:10.1090/S0002-9939-1994-1172954-7

Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series
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Proc. Amer. Math. Soc. 120 (1994), 421-424 Request permission

Abstract:

We prove that a series derived using Euler’s transformation provides the analytic continuation of $\zeta (s)$ for all complex $s \ne 1$. At negative integers the series becomes a finite sum whose value is given by an explicit formula for Bernoulli numbers.

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