Questions tagged [polylogarithms]

For questions about the polylogarithm function, which is a generalization of the natural logarithm.

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An identity for rational functions leading to equations for multiple polylogarithms

The following identity is not hard to prove: $$ \sum_{1\leq i_1<i_2<\ldots <i_{2n}\leq N} (-1)^{i_1+\ldots+i_{2n}}\frac{(1-x_{i_1})(1-x_{i_3})\ldots(1-x_{i_{2n-1}})}{(1-x_{i_2})(1-x_{i_4})\...
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201 views

What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...
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178 views

Polylogarithm : reference request for proof of integral representation

On page 494 of the book Integrals and series, volume I : elementary functions, Gordon and Breach, 1986, by A. P. Prudnikov, Y. A. Brychkov, and O.I. Marichev, (perhaps translated from Russian), the ...
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171 views

Simplify the difference of two dilogarithms--as in the logarithmic counterpart

This question--pertaining to the quantum-information-theoretic topic of "bound entanglement"--stems from the question and answer to https://math.stackexchange.com/questions/3464105/if-possible-...
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310 views

What is the current fastest method to calculate Lerch's Phi transcendent?

Lerch's Phi transcendent is $$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$ I am trying to compute this for the following parameters: $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (...
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131 views

Trigonometric cancellation on the unit circle

Let $z \in \mathbb{C}$ with $|z|=1$ and $z\ne 1$. Now consider the sum $$S(N,p)=\sum_{k=0}^N k^p z^k,$$ for some positive integers $N,p$. An immediate upper bound on $|S(N,p)|$ is $$|S(N,p)|\le C_1(...
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Limiting property of polylogarithm ratio

I try (without success) to figure out what could be the following limit if any... For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
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A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$. Mikhail Kurkov noticed that it ...
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154 views

Approximations of Polylogarithm and Lerch transcendent?

For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp ...
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546 views

Several conjectured identities for polylogarithms

I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured $$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{...
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1answer
275 views

Formula for negative polylogarithms

Theorem. We have that $\displaystyle \underbrace{x\left(\dfrac{d}{dx}\left(\cdots x \left(\dfrac{d}{dx} \left( \dfrac{x}{1-x}\right)\right)\cdots\right)\right)}_{\text{$x \frac{d}{dx}$ $m$ times}}=\...
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Root Polylogarithm Dominance Questions

Motivation: I am trying to work on a problem related to computing the roots of a certain family of polynomials related to integer partition theory. In particular, I have been trying to ``Bridge the ...
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102 views

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 $\eta(...
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145 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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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 $H(z)$...
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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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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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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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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 = \operatorname{Li_2(\alpha_1^{-630})}-2\operatorname{Li_2(\alpha_1^{-...
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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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839 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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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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119 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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Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z_1,z_2,z_3,z_4)$ parametrizing the tetrahedron in $\...
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289 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 f_3(1,...
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a dilogarithm identity: known or new?

I was playing around with dilogarithms and numerically found the following dilogarithm identity: $$\text{Li}_2\left(\frac{2 m}{m^2+m-\sqrt{((m-3) m+1) \left(m^2+m+1\right)}-1}\right)$$ $$-\text{Li}...
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525 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 why ...
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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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650 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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1answer
587 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 Z)\to\...
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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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590 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. ...