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

(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 closed-forms.


 

Recall that the Riemann zeta function,, is defined 

 (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 sine series converges to sine, namely
 
(Euler's product sine).
 
If we replace the variable x by ix, where , we then obtain
 
.
 
 
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 log of both sides of Euler’s product sine series.  We then take the derivative of both sides in associating with the given identities above to 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 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 re-verify the accuracy of these special values before usage.)


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 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.

 


In-Text or Website Citation
Tue N. Vu, A Brief Note - Sum of Riemann Zeta Function - Digamma Function, 06/14/2009, from Series Math Study Resource.

 A mathematical truth is neither simple nor complicated in itself, it is. (Emile Lemoine)