On the distribution of the values of short sums of Dirichlet characters over prime numbers. (Q1425537)

The author considers a sum of the form of \(S_h(\chi) = \sum\limits_{p\leq h} \chi(p)f(p)\), where \(p\) is a prime number, \(h\) is an integer, \(\chi\) is a Dirichlet character modulo \(m\) and \(f(n)\) is a complex-valued number-theoretic function of modulus one. The main result of the paper is that the random variable \(\xi_{\chi}=| S_ h(\chi)| ^2/\pi(h)\) has a distribution function as follows. Let \(h=h(m)\) depend on \(m\) in such a way that \(\lim_{m\to\infty}\log h(m)/\log m=0\) and \(\lim_{m\to\infty}h(m)=\infty\); then the inequality \(\xi_{\chi}\leq\lambda\) holds with a certain asymptotic probability as \(m\) tends to infinity.