1Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil
Turkish Journal of Analysis and Number Theory.
2017,
Vol. 5 No. 4, 143-145
DOI: 10.12691/tjant-5-4-5
Copyright © 2017 Science and Education PublishingCite this paper: F. M. S. Lima. An Elementary Proof of
and a Recurrence Formula for ζ(2
k).
Turkish Journal of Analysis and Number Theory. 2017; 5(4):143-145. doi: 10.12691/tjant-5-4-5.
Correspondence to: F. M. S. Lima, Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil. Email:
fabio@fis.unb.brAbstract
In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values ζ(2
k+1), ζ(
s) being the Riemann zeta function and
k a positive integer, is modified in a manner to furnish the even zeta values ζ(2
k). As a result, we find an elementary proof of
, as well as a recurrence formula for ζ(2
k) from which it follows that the ratio ζ(2
k)/
π2k is a rational number, without making use of Euler's formula and Bernoulli numbers.
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