Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 4 (2019), 541-548.

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

Samyukta Krishnamurthy and Micah B. Milinovich

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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 12.

Article information

Source
Involve, Volume 12, Number 4 (2019), 541-548.

Dates
Received: 12 June 2017
Revised: 30 July 2018
Accepted: 28 October 2018
First available in Project Euclid: 30 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1559181651

Digital Object Identifier
doi:10.2140/involve.2019.12.541

Mathematical Reviews number (MathSciNet)
MR3941597

Zentralblatt MATH identifier
07072538

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11B68: Bernoulli and Euler numbers and polynomials 11B37: Recurrences {For applications to special functions, see 33-XX}

Keywords
Riemann zeta function Euler Basel problem Bernoulli numbers Bernoulli polynomials

Citation

Krishnamurthy, Samyukta; Milinovich, Micah B. Euler's formula for the zeta function at the positive even integers. Involve 12 (2019), no. 4, 541--548. doi:10.2140/involve.2019.12.541. https://projecteuclid.org/euclid.involve/1559181651


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References

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