SUMS OF 2 K-TH POWERS - AN OMEGA-ESTIMATE FOR THE ERROR TERM

The arithmetic function r(k)(n) counts the number of ways to write a n atural number n as a sum of two k-th powers (k greater than or equal t o 2 fixed). The investigation of the asymptotic behaviour of the Diric hlet summatory function of rk(n) leads in a natural way to a certain e rror term P-k(t) In this article, we establish an Q-estimate for P-k(t ) (k > 2 arbitrary) which is essentially as sharp as the best known on e in the classic case k = 2.