LOG#160. Polylogia flashes(II).

IntegerRelationsPolylog1q

The polylogarithm or Jonquière’s function is generally defined as

    \[\boxed{Li_s(z)=\sum_{n=1}^\infty \dfrac{z^n}{n^s}}\]

Do not confuse Li_s(z) with the logarithm integral in number theory, which is

    \[li(x)=\int_0^x\dfrac{dt}{\log t}\]

such as

    \[\int_0^x\dfrac{dt}{\ln t}\]

\forall 0<x<1 and

    \[V.P.\int_0^x\dfrac{dt}{\ln t}=\lim_{\varepsilon\rightarrow{0^+}}\left[\int_0^{1-\varepsilon}\dfrac{dt}{\ln t}+\int_{1+\varepsilon}^x\dfrac{dt}{\ln t}\right]\]

\forall x>1. In fact, notation can be confusing sometimes since the european li(x) is sometimes written as Li(s) and

    \[\int_0^1li(z)dz=-\ln(2)\]

    \[Li(x)=\int_2^x=li(x)-li(2)\]

The polylogarithm (or polylog, for short) is sometimes written as F(z,s), i.e.,

    \[Li_s(z)=F(z,s)\]

so…Be aware with notations and the meaning of the symbols! The polylog is related with the Lerch trascendent in the following way

(1)   \begin{equation*} \boxed{Li_s(z)=z\Phi(z,s,1)} \end{equation*}

 The polylog is a wonderful function. It is ubiquitous in Physics, Chemistry and Mathematics…For instance, it arises in:

1st. Feynman diagram integrals, renormalization, and, in particular, in the calculation of the QED (Quantum Electrodynamics) corrections to the electron gyromagnetic ratio, supegravity amplitudes and other quantum scattering problems.

2nd. Quantum statistics. The Fermi-Dirac and the Bose-Einstein statistics can be written in terms of polylogarithms.

3rd. Vacuum effects in strong fields and quantum gravity. Non-perturbative effects in QFT (Schwinger effect, instanton effects and others) can be handled with these incredible functions.

Let us analyze the case of quantum statistics a little. The Fermi-Dirac statistics/distribution can be written as follows

(2)   \begin{equation*} \int_0^\infty \dfrac{k^s}{e^{k-\mu}+1}=-\Gamma(s+1)Li_{s+1}(-e^{\mu}) \end{equation*}

The Bose-Einstein statistics/distribution can be written as follows

(3)   \begin{equation*} \int_0^\infty\dfrac{k^s}{e^{k-\mu}-1}=\Gamma(s+1)Li_{s+1}(e^{\mu}) \end{equation*}

The polylog can  be easily related to the Riemann zeta function

    \[\boxed{Li_s(1)=\zeta(s)}\]

In fact, “colored” polylogs do exist. We will write on them later…

Remark: Li_4\left(\dfrac{1}{2}\right) does appear in the 3rd order correction to the gyromagnetic ratio of the electron in QED.

Polylogs have some really cool and interesting properties and values. Firstly, the polylog is itself a polylog when it is derived (so it has a striking similarity with the classical exponential function):

(4)   \begin{equation*} \dfrac{d}{dx}\left(Li_n(x)\right)=\dfrac{1}{x}Li_{n-1}(x) \end{equation*}

This self-similarity is certainly suggestive for certain equations of mathematical physics. Moreover, some surprising identities of the polylogarith are like this one (Bayley et al. proved it):

(5)   \begin{equation*} \begin{split} \dfrac{Li_m(1/64)}{6^{m-1}}-\dfrac{Li_m(1/8)}{3^{m-1}}-\dfrac{2Li_m(1/4)}{2^{m-1}}+\dfrac{4Li_m(1/2)}{9}-\\ -\dfrac{5(-\ln 2)^m}{9m!}+\dfrac{\pi^2(-\ln 2)^{m-2}}{54(m-2)!}-\dfrac{\pi^4(-\ln 2)^{m-4}}{486(m-4)!}-\\ -\dfrac{403\zeta (5)(-\ln 2)^{m-5}}{1296(m-5)!}=0 \end{split} \end{equation*}

 No algorithm is known yet for integration of polylogarithms of functions in closed form. By the other hand, polylogs can be also defined for negative values of “s” and integer numbers, i.e., for all n=-1,-2,-3,-4,-5,\ldots,-\infty

In fact, we have

    \[\boxed{Li_{-n}(z)=\sum_{k=1}^\infty k^nz^k=\dfrac{1}{(1-z)^{n+1}}\sum_{i=0}^n\langle\begin{matrix}n\\ i\end{matrix}\rangle z^{n-i}}\]

and where

    \[\langle\begin{matrix}n\\ i\end{matrix}\rangle\]

is an eulerian number.

Polylogs also arise in the theory of generalized harmonic numbers H_{n,q}

    \[\sum_{n=1}^\infty H_{n,q}z^n=\dfrac{Li_q(z)}{1-z}\]

with \vert z\vert<1. There are some special polylogs with special (perhaps subjective) beauty

    \[Li_{-2}(x)=\dfrac{x(x+1)}{(1-x)^3}\]

    \[Li_{-1}(x)=\dfrac{x}{(1-x)^2}\]

    \[Li_{0}(x)=\dfrac{x}{1-x}\]

Note that these are rational functions! We also have

    \[Li_1(x)=-\ln (1-x)\]

    \[Li_n(-1)=-\eta(n)\]

    \[Li_x(1)=\zeta(x)\]

    \[Li_n(1)=\zeta(n)\]

Some interesting known values of the polylog

    \[Li_1\left(\dfrac{1}{2}\right)=\ln (2)\]

    \[Li_2\left(\dfrac{1}{2}\right)=\dfrac{1}{12}\left(\pi^2-6(\ln 2)^2\right)\]

    \[Li_3\left(\dfrac{1}{2}\right)=\dfrac{1}{24}\left[4(\ln 2)^3-2\pi^2\ln 2+21\zeta (3)\right]\]

No higher formulas are known for Li_n(1/2) if n=4,5,\ldots

Euler’s dilogarithm is defined to be

    \[Li_2(z)=\sum_{k=1}^\infty\dfrac{z^k}{k^2}=-\int_0^z\dfrac{\ln (1-t)dt}{t}\]

It satisfies many functional identities and the dilogarithm values are interesting theirselves in many calculations. The trilogarithm or trilog can also be written to be

    \[Li_3(z)=\sum_{k=1}^\infty\dfrac{z^k}{k^3}\]

and where

    \[Li_3(-1)=-\dfrac{3}{4}\zeta (3)\]

and the beautiful result

(6)   \begin{equation*} \begin{split} Li_3(z)+Li_3(1-z)+Li_3(1-z^{-1})=\\ \zeta (3)+\dfrac{\ln^3(z)}{6}+\dfrac{\pi^2}{6}\ln(z)-\dfrac{1}{2}\ln^2 z\ln(1-z) \end{split} \end{equation*}

and thus we obtain another wonderful result

    \[\boxed{\zeta (3)=\dfrac{4}{7}\left[2Li_3(2)-\dfrac{\ln 2}{2}\pi^2\right]=\dfrac{8Li_3(2)-2\pi^2\ln 2}{7}}\]

The multidimensional polylog is defined as

(7)   \begin{equation*} Li_{s_1,\ldots,s_m}(z)=\sum_{n_1>\ldots>n_m>0}\dfrac{z^{n_1}}{n_1^{s_1}\cdots n_m^{s_m}} \end{equation*}

Furthermore, the colored polylog is

(8)   \begin{equation*} Li_{s_1,\ldots ,s_k}(z_1,\ldots,z_k)=\sum_{\substack{s_1,\ldots,s_k\\ n_1>\ldots>n_m>0}}\dfrac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}} \end{equation*}

The Nielsen generalized polylogarithm is

    \[S_{n-1,1}(z)=Li_n(z)\]

such as

    \[S_{n,p}(z)=\dfrac{(-1)^{n+p-1}}{(n-1)!p!}\int_0^1\dfrac{(\ln t)^{n-1}\left[\ln (1-zt)\right]^pdt}{t}\]

and it is computed as Polylog[n,p,z] sometimes. The Nielsen-Ramanujan constanst are also beautiful:

    \[a_k=\int_1^2\dfrac{\ln^k xdx}{x-1}\]

and

    \[a_p=p!\zeta (p+1)-\dfrac{p\left(\ln 2\right)^{p+1}}{p+1}-p!\sum_{k=0}^{p-1}\dfrac{Li_{p+1-k}(1/2)(\ln 2)^k}{k!}\]

with

a_1=\dfrac{\zeta (2)}{2}=\dfrac{\pi^2}{12}, a_2=\dfrac{\zeta (3)}{4},…

 Finally, we are going to define higher order prime zeta functions. Recall that the classical Riemann zeta function and the prime zeta function have already being defined as

    \[\zeta(s)\equiv \zeta_0 (s)=\sum_{n=1}^\infty n^{-s}\]

    \[\zeta_1(s)\equiv P(s)=\sum_{n=1}^\infty p_n^{-s}=\sum_{n=1}^\infty p(n)^{-s}\]

and thus, the k-th order prime zeta function should be written as

    \[\zeta_k(s)=\sum_{n=1}^\infty \left(P^{(k)}(n)\right)^{-s}=\sum_{n=1}^\infty\left(\underbrace{P(P(\cdots))}_\text{k-times}(n)\right)^{-s}\]

Moreover, the prime zeta function of infinite order should be \zeta_\infty (s)=1. The iterated \Omega-power of the Riemann zeta function is

    \[\underbrace{\zeta(\zeta(\cdots))}_\text{$\Omega$- times}=\zeta^{(\Omega(s))}\equiv \Omega_\zeta(s)\]

We could call it the omega zeta function.

See you in my next polylog post!!!!!!

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