All Questions
Tagged with polylogarithm zeta-functions
9
questions with no upvoted or accepted answers
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More on the log sine integral $\int_0^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\mathrm{d}\theta$
I. In this post, the OP asks about the particular log sine integral,
$$\mathrm{Ls}_{7}^{\left ( 3 \right )} =-\int_{0}^{\pi }\theta ^{3}\log^{3}\left ( 2\sin\frac{\theta }{2} \right )\,\mathrm{d}\...
- 54.9k
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Why does the tribonacci constant have a trilogarithm ladder?
When I came across the dilogarithm ladders of Coxeter and Landen, namely,
$$\text{Li}_2\Big(\frac1{\phi^6}\Big)-4\text{Li}_2\Big(\frac1{\phi^3}\Big)-3\text{Li}_2\Big(\frac1{\phi^2}\Big)+6\text{Li}_2\...
- 54.9k
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Patterns for the polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ and Nielsen polylogarithm $S_{n,p}\big(\tfrac12\big)$?
The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,...
- 54.9k
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Difference of polylogarithms of complex conjugate arguments
I have the expression
$$\tag{1}
\operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*)
$$
Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
- 4,817
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Upper bounds regarding polylogarithm $Li_p(e^w)$ when $|w| > 2\pi$ and p negative real
$\textit{The Computations of Polylogarithms, 1992,Technical report UKC, University of Kent, Canterbury, UK}\\ $ by $\textbf{David C Wood}$ says that
$$
Li_p(e^w) = \sum_{n\geq 0} \zeta(p-n)\frac{w^n}{...
- 417
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Relations between Dilogarithms and Imaginary part of Hurwitz-Zeta function
I'm working through a paper that involves a problem concerning the calculation of the Imaginary part of the derivative of the Hurwitz-Zeta function $\zeta_H(z,a)$ with respect to $z$, evaluated at a ...
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Further question on Logarithm product integral
How to perform $\int_0^1 \frac{\left(a_0\log(u)+a_1\log(1-u)+a_{2}\log(1-xu)\right)^9}{u-1} du $?
Method tried:
Intgration-by-parts
Series expansion
change of variable $\log(u)=x$
But I still can't ...
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How does $\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w})$ become $\Gamma(1-s)(-w)^{s-1}$?
I am trying to prove the following property of polylogarithm. $$\lim\limits_{Re(s)\rightarrow -\infty} Li_s(e^{w}) = \Gamma(1-s)(-w)^{s-1} \text{ for } -\pi < \Im(w) < \pi.$$
As per Wikipedia's "...
- 417
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Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?
This is a subquestion of this question on MO.
Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of:
$$Li_s(z)...
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