(June 14, 2009)
In this brief note we show how to derive a sum of Riemann zeta function [1*] in terms ofand Euler constants () with involving the Gamma function [2*].
.
This sum is also used as a connection to Digamma function [3*] in which some special values of Digamma function are computed in closedforms.
Recall that the Riemann zeta function,, is defined
 x = 1:
 x = 1:
 x = 1/2:
 x = 1, it gives
 x = 1/2, it gives
.
Digamma Function ()
The digamma function is defined as
. (II)
It is interested that by differentiating (I) and its result gives a connection to digamma function which satisfies the following expression:
. (III)
Special values
. (Notice that at x = 1 does not satisfy (III) except its definition (II).)
.
.
.
.
.
.
(Notice that we evaluate the special values ofand. Readers should reverify the accuracy of these special values before usage.)
Relation Series
 Series List V in relation to Euler's constant ().
References
[1*] Riemann zeta function
 Riemann zeta function, http://en.wikipedia.org/wiki/Riemann_zeta_function, from Wikipedia resource.
 Weisstein, Eric W. "Riemann Zeta Function." From Mathworld  A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannZetaFunction.html.
[2*] Gamma function

Gamma function, http://en.wikipedia.org/wiki/Gamma_function, from Wikipedia resource.

Weisstein, Eric W. "Gamma Function." From Mathworld  A Wolfram Web Resource.
[3*] Digamma function
 Digamma function, http://en.wikipedia.org/wiki/Gamma_function, from Wikipedia resource.

Weisstein, Eric W. "Digamma Function." From Mathworld  A Wolfram Web Resource.
(Note: the above links may be changed by other websites in future.)
More about Riemann, especially Riemann hypothesis, which is part of Problem 8 in Hilbert's list of 23 unsolved problems, and is one of the Clay Mathematics Institute Millennium Prize Problems.
A mathematical truth is neither simple nor complicated in itself, it is. (Emile Lemoine)
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