A remark on the gamma function (Q912887)

Let P denote the arithmetic function defined by \(P(n)=\prod_{k=1,...,n;(k,n)=1}\Gamma (k/n)\), where \(\Gamma\) is the Euler gamma function. The authors give an explicit formula for P(n) and establish an asymptotic formula with remainder term for the summatory function of log P(n). The motivation for the study of the function P arises from the relation log P(n)\(\sim \phi (n) \log \sqrt{2\pi}\) (n\(\to \infty)\), where \(\phi\) is the Euler totient function.