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Using a polylogarithmic identity, we express the values of $\zeta$ at oddintegers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simplefunctions involving products of logarithms. We also prove a useful property ofthose functions as some of their variables are raised to a power. In the case$n=2$, we prove two closed-form expressions concerning related integrals.Finally, another family of related one-dimensional integrals is studied.

A formula for $\zeta$(2n + 1) and some related expressions