Contour integration underlies fundamental Bernoulli number recurrence

TL;DR

This paper demonstrates how contour integration and log-sine evaluations can derive Bernoulli number recurrences and Euler's connection, linking complex analysis with classical number theory results.

Contribution

It introduces a novel contour integral approach to derive Bernoulli number recurrences and Euler's connection, providing alternative proofs and elementary interpretations.

Findings

Derived Bernoulli number recurrence via contour integrals.

Connected Bernoulli numbers to Riemann zeta function values.

Provided elementary Fourier series-based derivations of key integrals.

Abstract

One solution to a relatively recent American Mathematical Monthly problem [6], requesting the evaluation of a real definite integral, could be couched in terms of a contour integral which vanishes {\textit{a priori.}} While the required real integral emerged on setting to zero the real part of the contour quadrature, the obligatory, simultaneous vanishing of the imaginary part alluded to still another pair of real integrals forming the first two entries in the infinite log-sine sequence, known in its entirety. It turns out that identical reasoning, utilizing the same contour but a slightly different analytic function thereon, sufficed not only to evaluate that sequence anew, on the basis of a vanishing real part, but also, in setting to zero its conjugate imaginary part, to recover the fundamental Bernoulli number recurrence. The even order Bernoulli numbers $B_{2k}$ entering therein were revealed on the basis of their celebrated connection to Riemann's zeta function $\zeta(2k).$ Conversely, by permitting the related Bernoulli polynomials to participate as integrand factors, Euler's connection itself received an independent demonstration, accompanied once more by an elegant log-sine evaluation, alternative to that already given. And, while the Bernoulli recurrence is intended to enjoy here the pride of place, this note ends on a gloss wherein all the motivating real integrals are recovered yet again, and in quite elementary terms, from the Fourier series into which the Taylor development for Log$(1-z)$ blends when its argument $z$ is restricted to the unit circle.