(June 14, 2009)
In this brief note, we demonstrate how to derive the following sum that involves the Riemann zeta function [1*] in terms of π, Euler constants γ, and the Gamma function [2*].
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This sum is also connected to the Digamma function [3*], where certain special values of the Digamma function are computed in closed form.
Recall that the Riemann zeta function, denoted as, is a complex function that plays a crucial role in number theory and has applications in physics, probability theory, and applied statistics. It is defined in the integral form,
 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 differentiating (I) results in a connection to the digamma function, which satisfies the following expression:
. (III)
Special values
. (Notice that at x = 1, it does not satisfy (III) except for its definition in (II).)
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(Notice that we evaluate the special values ofand. Readers should reverify the accuracy of these special values before using them.)
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 the 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.
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