eng
Science and Education Publishing
Turkish Journal of Analysis and Number Theory
2333-1232
2017-06-27
5
4
143
145
10.12691/tjant-5-4-5
TJANT2017545
article
An Elementary Proof of and a Recurrence Formula for (2k)
F. M. S. Lima
fabio@fis.unb.br
1
Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil
In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values (2k+1), (s) being the Riemann zeta function and k a positive integer, is modified in a manner to furnish the even zeta values (2k). As a result, we find an elementary proof of , as well as a recurrence formula for (2k) from which it follows that the ratio (2k)/2k is a rational number, without making use of Euler's formula and Bernoulli numbers.
http://pubs.sciepub.com/tjant/5/4/5/tjant-5-4-5.pdf
Riemann zeta function
Euler��s formula
Euler polynomials