×

zbMATH — the first resource for mathematics

Euler’s formula for the zeta function at the positive even integers. (English) Zbl 1428.11147
Summary: 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 \(\frac{1}{2}\).
MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11B68 Bernoulli and Euler numbers and polynomials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 10.1090/S0002-9939-2010-10565-8 · Zbl 1223.40001
[2] 10.2307/2319041 · Zbl 0293.10001
[3] 10.2307/2690371 · Zbl 0526.01015
[4] ; Montgomery, Multiplicative number theory, I : Classical theory. Multiplicative number theory, I : Classical theory. Cambridge Studies in Advanced Mathematics, 97, (2007) · Zbl 1142.11001
[5] 10.1090/S0273-0979-07-01175-5 · Zbl 1135.01010
[6] 10.1007/978-0-8176-4571-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.