On a conjecture of Murty-Saradha about digamma values (Q2165632)

Let \(q>1\) be an integer and \(K\) be an algebraic number field over which the \(q\)-th cyclotomic polynomial is irreducible. Set \(\psi(x)=\frac d{dx} \log \Gamma(x)\). Then the authors prove that the numbers \(\psi(a/q)\), where \(1\leq a\leq q-1\) and \((a,q)=1\), are linearly independent over \(K\) with at most one exceptional \(q\). \\ From this we obtain some information about Murty-Saradha conjecture about digamma values.