LOG#159. Polylogia flashes (I).

Posted on by April 3, 2015

Complex_polylog2

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)   \begin{equation*} \zeta (s,a)=\sum_{n=0}^\infty \dfrac{1}{(a+n)^s}=\dfrac{1}{a^s}+\sum_{n=1}^\infty\dfrac{1}{(a+n)^s} \end{equation*}

2. k-th derivative for Hurwitz zeta function \zeta (s,a), Re(s)>1, s\neq 1, is:

(2)   \begin{equation*} \dfrac{d^k\zeta (s,a)}{ds}=(-1)^k\sum_{n=0}^\infty \dfrac{\log^k(a+n)}{(a+n)^s}\equiv \zeta^{(k)} (s,a) \end{equation*}

3. Dirichlet L-series or alternate zeta series:

(3)   \begin{equation*} \eta (s)=\sum_{n=0}^\infty\dfrac{(-1)^n}{n^s}=1-2^{-s}+3^{-s}-4^{-s}+\cdots \end{equation*}

with \eta (1)=\log (2) and

    \[\boxed{\eta^s=(1-2^{1-s})\zeta (s)}\]

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

(4)   \begin{equation*} L(s,z)=\sum_{n=1}^\infty \dfrac{\chi (n)}{n^s} \end{equation*}

More interestingly, there are some interesting constants worth studying. They are called Stieltjes constants. \gamma_n (Stieltjes constants) are the n-th coefficients in the Laurent series expansion of the Riemann zeta function at s=1 (the pole!). That is:

(5)   \begin{equation*} \zeta (s)=\dfrac{1}{s-1}+\sum_{n=0}^\infty \dfrac{(-1)^n}{n!}\gamma_n (s-1)^n \end{equation*}

and where

    \[\gamma_n=\lim_{m\rightarrow\infty}\left[\left(\sum_{k=1}^m\dfrac{(\ln k)^n}{k}\right)-\dfrac{(\ln m)^{n+1}}{n+1}\right]\]

Indeed, the 0-th Stieltjes constant is usually called the Euler-Mascheroni constant \gamma_E=\gamma_0. More generally, we can even define the Stieltjes constants for the Hurwitz zeta function, and we will denote it by \gamma_n (a). Thus, we have:

(6)   \begin{equation*} \gamma_E=\gamma_0=\lim_{n\rightarrow \infty}\left( \sum_{k=1}^n \dfrac{1}{k}-\ln n \right) \end{equation*}

or

(7)   \begin{equation*} \gamma_E=\sum_{n=1}^\infty\left[\dfrac{1}{k}-\ln\left(1+\dfrac{1}{k}\right)\right] \end{equation*}

(8)   \begin{equation*} \gamma_E=\sum_{m=2}^\infty(-1)^m\dfrac{\zeta(m)}{m} \end{equation*}

and

(9)   \begin{equation*} \zeta(s,q)=\dfrac{1}{s-1}+\sum_{n=0}^\infty\dfrac{(-1)^n}{n!}\gamma_n(q)(s-1)^n \end{equation*}

They satisfy as well

(10)   \begin{equation*} \gamma_n(a) =\lim_{m\to\infty}\left[ \sum_{k=0}^m \frac{\ln^n (k+a)}{k+a} - \frac{\ln^{n+1} (m+a)}{n+1} \right] \end{equation*}

for n=0,1,2,\ldots and a\neq 0,-1,-2,\ldots

5. Siegel zeta function. It is defined as follows

(11)   \begin{equation*} Z(t)=e^{i\theta (t)}\zeta (\dfrac{1}{2}+it) \end{equation*}

where \theta(t) is the Riemann-Siegel theta function. It is defined as

(12)   \begin{equation*} \theta(t)=\dfrac{\log \Gamma \left(\dfrac{1+2it}{4}\right)-\log \left(\dfrac{1-2it}{4}\right)}{2i}-\dfrac{\log \pi}{2}t \end{equation*}

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

6. Lerch trascendent.

(13)   \begin{equation*} \Psi(z,s,a)=\sum_{k=0}^\infty \dfrac{z^n}{(a+k)^s} \end{equation*}

(14)   \begin{equation*} \Psi(z,s,a)=z\Psi(z,s,a+1)+a^{-s} \end{equation*}

(15)   \begin{equation*} \Psi(z,s,a)=\dfrac{1}{2a^s}+\int_0^\infty\dfrac{z^t}{(a+t)^s}dt-2\int_0^\infty \dfrac{\sin (t\log z-s\arctan(\frac{t}{a}))}{(a^2+t^2)^{s/2}(e^{2\pi t}-1)}dt \end{equation*}

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

7. The polylogarithm.

(16)   \begin{equation*} Li_s(z)=\sum_{k=1}^\infty\dfrac{z^k}{k^s} \end{equation*}

8. Clausen functions. They are defined as

(17)   \begin{equation*} Cl_s(z)=\sum_{k=1}^\infty \dfrac{\sin (kz)}{k^s} \end{equation*}

Clausen functions are classically defined as Cl_2(z), but Cl_s(z) 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)   \begin{equation*} Cl_s(z)=\dfrac{1}{2i}\left( Li_s(e^{iz})-Li_s(e^{-iz})\right) \end{equation*}

so

    \[\boxed{Cl_s(z)=Im(Li_s(e^{iz}))}\]

\forall s,z\in R or \in C by analytic continuation.  Moreover, we have the Kummer’s relation

    \[Li_2(e^{i\theta})=\zeta (2)-\dfrac{\theta(2\pi-\theta)}{4}+iCl_2(\theta)\]

and

    \[Cl_2(\dfrac{\pi}{2})=G\]

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)   \begin{equation*} Cl_n(x)=\begin{cases}S_n(x)=\displaystyle{\sum_{k=1}^\infty} \dfrac{\sin(kx)}{k^n}\;\;\mbox{n even}\\ C_n(x)=\displaystyle{\sum_{k=1}^\infty} \dfrac{\cos{kx}}{k^n}\;\; \mbox{n odd}\end{cases} \end{equation*}

(20)   \begin{equation*} Cl_n(x)=\begin{cases}S_n(x)=\dfrac{1}{2i}\left[Li_n(e^{ix})-Li_n(e^{-ix})\right]\;\;\mbox{n even}\\ C_n(x)=\dfrac{1}{2}\left[Li_n(e^{ix})-Li_n(e^{-ix})\right]\;\; \mbox{n odd}\end{cases} \end{equation*}

and

    \[Cl_1(x)=s_1(x)=-\ln\vert 2\sin\dfrac{x}{2}\vert\]

    \[Cl_2(x)=s_2(x)=-\int_0^x\ln\left[2\left(\sin(\dfrac{t}{2}\right)\right]dt\]

For Clausen cosine functions \tilde{Cl}_s(z) we have

(21)   \begin{equation*} \tilde{Cl}_s(z)=\sum_{k=1}\dfrac{\cos(kz)}{k^s} \end{equation*}

(22)   \begin{equation*} \tilde{Cl}_s(z)=\dfrac{1}{2}\left[Li_s(e^{iz})+Li_s(e^{-iz})\right]=Re\left[Li_s(e^{iz})\right] \end{equation*}

 9. Polyexponential.

(23)   \begin{equation*} E_s(z)=\sum_{k=1}^\infty\dfrac{k^s}{k!}z^k=\sum_{k=1}^\infty\dfrac{k^s}{\Gamma(k+1)}z^k=\sum_{m=2}^\infty\dfrac{(m-1)^s}{\Gamma (m)}z^{m-1} \end{equation*}

where m=k+1 and E_s(z)=e^zB_s(z)-\mbox{sinc} (\pi s), B_s(z) are the Bell polymials and \mbox{sinc} is the circular sine function.

10. Prime zeta and second zeta.

(24)   \begin{equation*} P(s)=\sum_{prime}\dfrac{1}{p^s} \end{equation*}

(25)   \begin{equation*} Z(s)=\sum_{n=1}^\infty\dfrac{1}{\tau_n^s} \end{equation*}

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

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

(26)   \begin{equation*} k^m\Psi^{(m-1)}(kz)=\sum_{n=0}^{k-1}\Psi^{(m-1)}\left(z+\frac{n}{k}\right)\forall m>1 \end{equation*}

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

(27)   \begin{equation*} k\left[\Psi(kz)-\log(k)\right]=\sum_{n=0}^{k-1}\Psi (z+\frac{n}{k}) \end{equation*}

12. Gamma function-Legendre function.

Legendre gives

(28)   \begin{equation*} \Gamma (z)\Gamma (z+\frac{1}{2})=2^{1-2z}\sqrt{\pi}\Gamma (2z) \end{equation*}

and Gauss

(29)   \begin{equation*} \Gamma (z)\Gamma (z+\frac{1}{k})\Gamma(z+\frac{2}{k}\cdots \Gamma(z+\frac{k-1}{k}=(2\pi)^{(k-1)/2}k^{1/2-kz}\Gamma (kz) \end{equation*}

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

    \[k^{1-m}B_m(kx)=\sum_{n=0}^{k-1}B_m(x+\frac{n}{k})\]

and the Euler polynomials are given by

    \[k^{-m}E_m(kx)=\sum_{n=0}^{k-1}(-1)^nE_m(x+\frac{n}{k})\]

for k=1,2,… and

    \[k^{-m}E_m(kx)=-\dfrac{2}{m+1}\sum_{n=0}^{k-1}(-1)^nB_{m+1}(x+\frac{n}{k})\]

for k=2,4,…

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

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