A short account of the values of the zeta function at integers

Abstract

We use methods of real analysis to continue the Riemann zeta function $\zeta(s)$ to all complex $s$, and to express the values at integers in terms of Bernoulli numbers, using only those infinite series for which we could write down an explicit estimate for the remainder after $N$ terms. This paper is self-contained, apart from appeals to the uniqueness theorems for analytic continuation and for real power series, and, verbis in Latinam translatis, would be accessible to Euler.