On a conjecture of S. Chowla and of S. Chowla and H. Walum, I

Abstract

As an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, as x → ∞, Σn ≤ √x naBr({ x n}) = O(xa 2 + 1 4 + ε) holds for each ε > 0. Here integers a ≥ 0 and r ≥ 1 are given. Br(x) denotes the rth Bernoulli polynomial and {x} denotes the fractional part of x. The special case a = 0, r = 2 of this conjecture was also mentioned by S. Chowla. In this paper we prove this conjecture for all e ≥ 1 2 and r ≥ 2 with ε = 0 (with xε replaced by log x in case a = 1 2). © 1985.