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 $\frac{1}{2}$.

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Keywords
Riemann zeta function, Euler, Basel problem, Bernoulli numbers, Bernoulli polynomials
Mathematical Subject Classification 2010
Primary: 11M06
Secondary: 11B68, 11B37