Number Theory

    

Lerch's Φ and the Polylogarithm at the Positive Integers

Authors: Jose Risomar Sousa

We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) sim H(n)-sum_{kge 2}(-1)^kzeta(k)b^{k-1}$). Finally, we use the found Fourier series formulae to obtain the values of the Lerch transcendent function, $Phi(e^z,k,b)$, and by extension the polylogarithm, $mathrm{Li}_{k}(e^{z})$, at the positive integers $k$.

Comments: 17 Pages.

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Submission history

[v1] 2021-11-25 21:45:08
[v2] 2025-12-01 15:53:58
[v3] 2026-02-22 17:30:24

Unique-IP document downloads: 296 times

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