Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
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Turkish Journal of Analysis and Number Theory. 2017, 5(4), 143-145
DOI: 10.12691/tjant-5-4-5
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An Elementary Proof of and a Recurrence Formula for ζ(2k)

F. M. S. Lima1,

1Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil

Pub. Date: June 27, 2017

Cite this paper:
F. M. S. Lima. An Elementary Proof of and a Recurrence Formula for ζ(2k). Turkish Journal of Analysis and Number Theory. 2017; 5(4):143-145. doi: 10.12691/tjant-5-4-5


In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values ζ(2k+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(2k). As a result, we find an elementary proof of , as well as a recurrence formula for ζ(2k) from which it follows that the ratio ζ(2k)/π2k is a rational number, without making use of Euler's formula and Bernoulli numbers.

Riemann zeta function Euler’s formula Euler polynomials

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[1]  M. Aigner and G.M. Ziegler, Proofs from THE BOOK, 5th ed., Springer, New York, 2014, Chap. 9.
[2]  R. Ayoub, “Euler and the zeta function,” Am. Math. Monthly 81, 1067-1085 (1974).
[3]  M. J. Dancs and T.-X. He, “An Euler-type formula for ζ(2k+1),” J. Number Theory 118, 192-199 (2006).
[4]  L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740).
[5]  D. Kalman, “Six ways to sum a series,” Coll. Math. J. 24, 402-421 (1993).