Evaluation of Harmonic Sums with Integrals
Vivek Kaushik, Daniele Ritelli

TL;DR
This paper uses multiple integration techniques to evaluate harmonic sums and Riemann zeta values, revealing connections with probability distributions and geometric volumes, and generalizing classical identities.
Contribution
It introduces novel integral representations and probabilistic interpretations for harmonic sums and zeta values, extending known results to higher dimensions.
Findings
Derived new closed-form formulas for S(k) and ζ(2k).
Connected harmonic sums to volumes of convex polytopes.
Linked integrals to probabilities involving Cauchy and uniform random variables.
Abstract
We consider the sums and with being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show which implies Euler's identity Then, we generalize each integral in order to find the considered sums. The dimensional analogue of the first integral is the density function of the quotient of independent, nonnegative Cauchy random variables. In seeking this function, we encounter a special logarithmic integral that we can directly relate to The dimensional analogue of the second integral, upon a change of variables, is the volume of a convex polytope, which can be expressed as a probability…
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