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A Brief Note - Sum of Riemann Zeta Function - Digamma Function

Submitted by admin on Mon, 06/15/2009 - 1:59pm

(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*].

.

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,

 (Riemann’s integral form),
 
where  is the Gamma function.
 
 
Euler was the first person to define the Gamma function as a definite integral form
 
             (Euler’s integral form).
 
 
From the Euler’s integral form, it is not difficult to show Gamma function to satisfy the following identities:
 
.
 
            .
 
 
It is known that Euler’s product for sine converges to sine, namely
 
(Euler's product for sine).
 
Replacing x with ix gives
 
.
 
 
The Gamma function also satisfies the functional equation
 
.
 
Therefore, we can express the sum of Riemann zeta function in terms of and Euler constants with involving Gamma function by taking the logarithm of both sides of Euler’s product for sine.  By differentiating both sides and using the identities mentioned above, we obtain the following beautiful series:
 
,
 
where  is Euler's constant and its value is 0.5772156649015328606… .
 
 
Some special values
 
  • x = -1:
 
.
 
 
  • x = 1:
 
.
 
 
  • x = 1/2:
 
.
 
 
If we set x = -x in (I), we obtain the alternative series
 
.
 
Special values
 
  • 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).)

 

.

 

.

 

.

 

 

.

 

.

 

.

(Notice that we evaluate the special values ofand. Readers should re-verify the accuracy of these special values before using them.)

Relation Series

 

References

[1*] Riemann zeta function

 

[2*] Gamma function

 

[3*] Digamma function

(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.