Sums of two k-th powers: an Omega estimate for the error term

The arithmetic function r _k (n) counts the number of ways to write a natural number n as a sum of two k-th powers (k ≧ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of r _k(n) leads in a natural way to a certain error term P _k(t). In this article, we establish an Ω-estimate for P _k(t) (k τ; 2 arbitrary) which is essentially as sharp as the best known one in the classic case k=2. This article is part of a research project supported by the Austrian Science Foundation (Nr. P 9892-PHY).