On a conjecture of Chowla and Walum

Abstract

We show the asymptotic behaviour of the mean square of the sum \( \sum\nolimits_{n \leqslant c\sqrt x } {n^a P_k \left( {{x \mathord{\left/ {\vphantom {x n}} \right. \kern-\nulldelimiterspace} n}} \right)} \), where P k (x) = B k ({x}) and B k (x) denotes the Bernoulli polynomial of degree k and c > 0 is a real number such that c 2 is rational. Our result implies that a conjecture of Chowla and Walum is true on average.