The polylogarithms tag has no wiki summary.

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### problem solving logarithmic equation and reaching an equivalence [on hold]

ok so i've had a problem trying to simplify the ln[ sqrt(1+(u^2/a^2)) + u/a ] and this is supposed to be equal to : ...

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### Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms:
$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$
Assume $z=i$:
$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$
with ...

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69 views

### Are all complex zeros of $Li_s(z)\, \pm \, Li_{1-s}(z)$ on the critical line or outside the critical strip for $z \le -1$?

This question loosely builds on this one, however is a bit simpler and I found the results to be more robust.
It seems that all zeros in the critical strip $0 \lt \Re(s) < 1$ of:
$$Li_s(z)\, \pm ...

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254 views

### Dilogarithm of -1/2?

$\mathrm{Li}_2(\frac12)=\frac{\pi^2}{12}-\frac{\log^2(2)}{2}$ (L. Euler, 1768)
$\mathrm{Li}_2(-\frac12)=-\pi\arg\left(\frac{H(-1-\frac{\log(2)}{\pi i})}{H(\frac{\log(2)}{\pi i})}\right)$,
where ...

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103 views

### Higher Discrete logarithms over finite fields

The polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the ...

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90 views

### Inverse of Polylogarithm

I am considering the polylogarithm $Li_n(x)$
What is the inverse function for polylogarithm $Li_n(x)$, where n is any complex value?
Thanks,
Gevorg.

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241 views

### New formula for polylogarithm and the fastest converging series for $\zeta(3)$?

Recently I found a formula for the infinite hyperbolic sine and cosine series also described in a .pdf here.
I write them as polylogarithms, as they are already extended by the process of analytic ...

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204 views

### The relationship between the dilogarithm and the golden ratio

Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 ...

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101 views

### A second polylogarithm ladder for the tribonacci and n-nacci constants

In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with,
$$0 = ...

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214 views

### May a globally bounded G-function have a logarithmic branching? (On a conjecture of Ruzsa)

By a "globally bounded $G$-function," following G. Christol, I will mean a solution to a (minimal) linear differential equation on $\mathbb{P}^1$ (then necessarily of the Fuchsian type and with ...

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321 views

### Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...

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63 views

### getting specific function as sum of squares of sines

The series $G_s(x):=\sum_{n=1}^\infty n^{-s}sin^2(nx)$ is, up to a constant factor, equal to $Li_s(1)-\Re Li_s(e^{2ix})$, where $Li_s(\cdot)$ is a polylogarithm function. $G_2(x)\sim x$ for $x\ll 1$, ...

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69 views

### linear independence of values of the polylogarithm at different roots of unity

I am interested in the real and imaginary part of the complex polylogarithm
$$L_{k+1}(\zeta):=Re(\frac 1 {i^k}\sum_{m=1}^\infty \frac{\zeta^m}{m^{k+1}}),$$
where $\zeta$ is a primitive $n$-th root ...

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399 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

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238 views

### Are there any known bounds on this function?

For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show
If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re ...

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842 views

### a dilogarithm identity: known or new?

I was playing around with dilogarithms and numerically found the following dilogarithm identity:
$$\text{Li}_2\left(\frac{2 m}{m^2+m-\sqrt{((m-3) m+1)
\left(m^2+m+1\right)}-1}\right)$$
...

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327 views

### Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups?
This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain ...

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533 views

### Why is there a unique hyperbolic simplex of largest area?

Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...

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446 views

### Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...

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347 views

### Why does the Cheeger--Chern--Simons class descend to H_3(G/B)?

Cheeger and Simons here defined a family of characteristic classes for principal $G$-bundles ($G$ a Lie group). I'm interested in the special case of $\hat c_2:H_3(SL(2,\mathbb C);\mathbb ...

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### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

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426 views

### Polylogarithm inequality

Recall the polylogarithm $Li_s(z)=\sum_{n=1}^\infty \frac{z^n}{n^s}.$
For $|z|<1,$ is $\Re \left( \frac{Li_1(z)}{Li_2(z)} - \frac{2}{3}\frac{Li_2(z)}{Li_3(z)} \right)>0?$
The numerics suggest ...

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### What is special about polylogarithms that leads to so many interesting identities and applications?

I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. ...