An asymptotic formula for the \(2k\)-th power mean value of \(|(L^{\prime}/L)(1+it_0, \chi)|\)
Let \(q\) be a positive integer \(\geq 2\), \(\chi\) be a Dirichlet character modulo \(q\), \(L(s,\chi)\) be the attached Dirichlet \(L\)-function, and let \(L'(s,\chi)\) denote its derivative with respect to the complex variable \(s\). Let \(t_0\) be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the \(2k\)-th power mean value of the logarithmic derivative of \(L(s, \chi)\) at any point \(s=1+i t_0\) on the line \(Re\; s=1\) when χ runs over all Dirichlet characters modulo \(q\) (except the principal character when \(t_0=0\)). The estimates are given with some explicit error term which tends to \(0\) as \(q \to \infty\) while \(t_0\) is fixed. As a consequence of their result, the authors showed that the values of \(|\frac{L'}{L}(1+it_0,\chi)|^2\) behave according to some distribution law.
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