Define harmonic numbers for a complex argument $z$ as $H\_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$.


For $n\in\mathbb{N}$, $H\_n$ are usual harmonic numbers $\sum^n\_{k=1} k^{-1}$ . They are obviously rational and are known (Taeisinger 1915) to be non-integers for $n>1$.


*Question:* Is there a non-integer rational $q$ such that $H\_q\in\mathbb{Q}$?



Can a harmonic number be a rational number for non-integer rational argument?
The answer is "no". Your function $H\_z$ which is the same as $\psi(z+1)+\gamma$, where $\psi$ is the [digamma function](http://en.wikipedia.org/wiki/Digamma_function), and $\gamma$ is the Euler-Mascheroni constant, takes transcendental values at non-integer rationals. This is a theorem of M. Ram Murty and N. Saradha, ["Transcendental values of the digamma function"](http://www.sciencedirect.com/science/article/pii/S0022314X06002927).


Notice that at rational values the digamma function has an explicit evaluation given by [Gauss's formula](http://en.wikipedia.org/wiki/Digamma_function#Gauss.27s_digamma_theorem).


