LOG#159. Polylogia flashes (I).

In the next series of post, I am going to define (again) and write some cool identities of several objects that mathematicians and physicists know as zeta functions and polylogarithms. I have talked about them already here http://www.thespectrumofriemannium.com/2012/11/07/log051-zeta-zoology/

1. We define the Hurwitz zeta function as:

(1)

2. k-th derivative for Hurwitz zeta function , , , is:

(2)

3. Dirichlet L-series or alternate zeta series:

(3)

with and

4. Dirichlet L-function, with being a periodic sequence of length q or a Dirichlet character:

(4)

More interestingly, there are some interesting constants worth studying. They are called Stieltjes constants. (Stieltjes constants) are the n-th coefficients in the Laurent series expansion of the Riemann zeta function at (the pole!). That is:

(5)

and where

Indeed, the 0-th Stieltjes constant is usually called the Euler-Mascheroni constant . More generally, we can even define the Stieltjes constants for the Hurwitz zeta function, and we will denote it by . Thus, we have:

(6)

or

(7)

(8)

and

(9)

They satisfy as well

(10)

for and

5. Siegel zeta function. It is defined as follows

(11)

where is the Riemann-Siegel theta function. It is defined as

(12)

The Riemann-Stieltjes theta function gives the phase factor for Z(t).

6. Lerch trascendent.

(13)

(14)

(15)

The Lerch trascendent generalizes the Hurwitz zeta function and the polylogarithm. What is a polylogarithm? You will see it now…

7. The polylogarithm.

(16)

8. Clausen functions. They are defined as

(17)

Clausen functions are classically defined as , but is much more general. It is defined for general s and z, both complex numbers and even if the series itself does not converge! The Clausen sine functions (Clausen cosine functions can also be defined) are related to the polylogarithm:

(18)

so

or by analytic continuation.  Moreover, we have the Kummer’s relation

and

where G is the Catalan’s constant. The Clausen sine function (clsin(z)) and the Clausen cosine function (clcos(z)) satisfy a complementarity relationship:

(19)

(20)

and

For Clausen cosine functions we have

(21)

(22)

9. Polyexponential.

(23)

where and , are the Bell polymials and is the circular sine function.

10. Prime zeta and second zeta.

(24)

(25)

where are the imaginary part of the non-trivial zeroes for the Riemann zeta function.

11. Polygamma function (or log-derivative of the gamma function )

(26)

If m=1, we have the digamma function. For m>1, we have polygammas.

(27)

12. Gamma function-Legendre function.

Legendre gives

(28)

and Gauss

(29)

Finally, in connection with all this stuff, there are some additional polynomials. The Bernoulli polynomials are defined as

and the Euler polynomials are given by

for k=1,2,… and

for k=2,4,…

I will see you in my next blog post!!!

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