Series with summands involving harmonic numbers
Zhi-Wei Sun
TL;DR
The paper studies series built from harmonic numbers of order up to four in combination with binomial-sum terms, proposing 70 conjectures across four sections and supporting them with extensive computer verification. The author connects these sums to classical constants and functions such as $\zeta$, the Catalan constant $G$, Dirichlet $L$-values, and Bernoulli/Euler polynomials, and explores rich $p$-adic congruences and Fermat-quotient phenomena. Key contributions include exact evaluations linking zeta-values and $G$, numerous congruences modulo primes involving $B_{p-3}$ and $E_{p-3}$, and a broad general conjecture predicting $\log|m|/\pi$-type behavior for Ramanujan-type series. The work significantly expands the catalog of rapid-convergence series and arithmetic properties associated with harmonic-number sums, offering a wide array of targets for future rigorous proofs and applications in analytic and computational number theory.
Abstract
For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum_{k=0}^\infty(6k+1)\frac{\binom{2k}k^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}H_{k}^{(3)}\right) =\frac{25ζ(3)}{8π}-G,$$ where $G$ denotes the Catalan constant $\sum_{k=0}^\infty(-1)^k/(2k+1)^2$. This paper contains $70$ conjectures posed by the author during 2022--2023.