Cent. Eur. J. Math. • 8(3) • 2010 • 488-499
DOI: 10.2478/s11533-010-0030-7
Central European Journal of Mathematics
An accurate approximation
of zeta-generalized-Euler-constant functions
Research Article
Vito Lampret
1∗
1 University of Ljubljana, Ljubljana, Slovenia
Received 22 October 2009; accepted 17 March 2010
Abstract: Zeta-generalized-Euler-constant functions,
γ(s):=
∞
k=1
1
k
s
−
k+1
k
d x
x
s
and
γ(s):=
∞
k=1
(−1)
k+1
1
k
s
−
k+1
k
d x
x
s
,
defined on the closed interval [0,∞), where γ(1) is the Euler-Mascheroni constant and γ(1) = ln
4
π
, are studied
and estimated with high accuracy.
MSC: 33E20, 33F05, 11Y60, 40A05, 40A25, 65B15
Keywords: Alternating • Convergence acceleration • Estimate • Generalized-Euler-constant-function • Inequality • Series •
Zeta
© Versita Sp. z o.o.
1. Introduction
Quite a few generalizations of Euler-Mascheroni constant C has been studied recently [5–8]. The constant C and its
alternating version ln
4
π
can be expressed in terms of infinite series and double integrals [7]
C =
∞
k=1
1
k
− ln
k +1
k
=
[0,1]×[0,1]
1 −x
(1 − xy)(− ln xy)
d x d y (1)
ln
4
π
=
∞
k=1
(−1)
k−1
1
k
− ln
k +1
k
=
[0,1]×[0,1]
1 −x
(1+ xy)(− ln xy)
d x d y. (2)
∗
E-mail: vito.lampret@fgg.uni-lj.si
488