This problem is considered in russian forum

Juna (or someone else), would you mind translating the solution for us?

Actually, I don't think that this problem was solved in that topic,at least I didn't feel that it was after reading that topic . They are manly concerned with calculating the sum \[S(m;1)\] and I don't think that they are done even with that work :razz: .

Peter wrote: Juna (or someone else), would you mind translating the solution for us? Try so Tiks wrote: Actually, I don't think that this problem was solved in that topic,at least I didn't feel that it was after reading that topic I did not speak that problem is solved completely. But is there proved that amount transcendental if $k|m$

The question is equivalent to proving that the expression $\gamma +\psi(x)$ is irrational for $x\in\mathbb{Q}-\mathbb{N}$ where $\gamma$ is the euler-mascheroni constant and $\psi$ is the digamma function. Just refer a paper by M.Ram Murty and N. Saradha that proves that in this case they must in fact be transcendental. $\square$