Vol. 12, No. 4, 2019

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Euler's formula for the zeta function at the positive even integers

Samyukta Krishnamurthy and Micah B. Milinovich

Vol. 12 (2019), No. 4, 541–548
DOI: 10.2140/involve.2019.12.541
Abstract

We give a new proof of Euler’s formula for the values of the Riemann zeta function at the positive even integers. The proof involves estimating a certain integral of elementary functions two different ways and using a recurrence relation for the Bernoulli polynomials evaluated at 1 2.

Keywords
Riemann zeta function, Euler, Basel problem, Bernoulli numbers, Bernoulli polynomials
Mathematical Subject Classification 2010
Primary: 11M06
Secondary: 11B68, 11B37
Milestones
Received: 12 June 2017
Revised: 30 July 2018
Accepted: 28 October 2018
Published: 16 April 2019

Communicated by Filip Saidak
Authors
Samyukta Krishnamurthy
Department of Physics
University of Mississippi
University, MS
United States
Department of Physics
University of Massachusetts
Amherst, MA
United States
Micah B. Milinovich
Department of Mathematics
University of Mississippi
University, MS
United States