  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 three.
For example, we conjecture that
$$\sum_{k=0}^\infty(6k+1)\frac{\binom{2k}k^3}{256^k}\left(H_{2k}^{(3)}-\frac{7}{64}H_{k}^{(3)}\right)
  =\frac{25\zeta(3)}{8\pi}-G,$$
  where $G$ denotes the Catalan constant $\sum_{k=0}^\infty(-1)^k/(2k+1)^2$.
This paper contains $66$ conjectures posed by the author since October 2022.

Number Theory