Finite-part integral representation of the Riemann zeta function at odd positive integers and consequent representations

The values of the Riemann zeta function at odd positive integers, $ζ(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$. Integral representations for $ζ(2n+1)$ are then deduced from the finite-part integral representation. Certain relations between $ζ(2n+1)$ and $ζ'(2n+1)$ are likewise deduced, from which integral representations for $ζ'(2n+1)$ are obtained.

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