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:
2. k-th derivative for Hurwitz zeta function , , , is:
3. Dirichlet L-series or alternate zeta series:
4. Dirichlet L-function, with being a periodic sequence of length q or a Dirichlet character:
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:
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:
They satisfy as well
5. Siegel zeta function. It is defined as follows
where is the Riemann-Siegel theta function. It is defined as
The Riemann-Stieltjes theta function gives the phase factor for Z(t).
6. Lerch trascendent.
The Lerch trascendent generalizes the Hurwitz zeta function and the polylogarithm. What is a polylogarithm? You will see it now…
7. The polylogarithm.
8. Clausen functions. They are defined as
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:
or by analytic continuation. Moreover, we have the Kummer’s relation
where G is the Catalan’s constant. The Clausen sine function (clsin(z)) and the Clausen cosine function (clcos(z)) satisfy a complementarity relationship:
For Clausen cosine functions we have
where and , are the Bell polymials and is the circular sine function.
10. Prime zeta and second zeta.
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 )
If m=1, we have the digamma function. For m>1, we have polygammas.
12. Gamma function-Legendre function.
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
I will see you in my next blog post!!!