# Integrals of logarithmic functions and alternating multiple zeta values

In: Mathematica Slovaca, vol. 69, no. 2
Ce Xu

## Details:

Year, pages: 2019, 339 - 356
Keywords:
multiple zeta values, multiple polylogarithms, harmonic numbers, Euler sums
By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form $ζ ({\bar 1,{{\{1\}}m - 1},\bar 1,{{\{1\}}k - 1}}), (k,m\in \Bbb N)$ for $m=1$ or $k=1$, and $ζ ({\bar 1,{{\{1\}}m - 1},p,{{\{1\}}k - 1}}), (k,m\in\Bbb N)$ for $p=1$ and $2$, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and $\ln 2$. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at $((1) / (2))$.