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Martin Sleziak
Martin Sleziak
1:39 PM
new-tag A new tag multiple-zeta-functions was created by pisco. The same user created a tag-excerpt.
> Questions on the multiple zeta functions and its generalization: multiple polylogarithm, generalized polylogarithm, colored polylogarithm; their special values; and their algebraic-geometric aspects.
Multiple zeta function
In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined by ζ ( s 1 , … , s k ) = ∑ n 1 > n 2 > ⋯ >...
3
Q: A bizarre logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x} dx$

piscoTwo years ago, I discovered the following neat result:$$\tag{*}\small{ \int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x} dx = -168 \text{Li}_5\left(\frac{1}{2}\right) \zeta (3)+96 \text{Li}_4\left(\frac{1}{2}\right){}^2-\frac{19}{15} \pi ^4 \text{Li}_4\left(\frac{1}{2}\right)+\\ 12 \pi ^2 \tex...

integration sequences-and-series zeta-functions polylogarithm multiple-zeta-functions
 

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