ACTA ARITHMETICA XCIX.2 (2001) On sums of two kth powers: a mean-square asymptotics over short intervals by Manfred K¨ uhleitner and Werner Georg Nowak (Wien) 1. Introduction. For k ≥ 2 a fixed integer, define the arithmetic func- tion r k (n) as the number of ways to write n ∈ N ∗ as a sum of two kth powers of absolute values of integers, i.e., r k (n)=#{(u 1 ,u 2 ) ∈ Z 2 : |u 1 | k + |u 2 | k = n}. To describe its average behaviour, one is interested in asymptotic results about the Dirichlet summatory function R k (u)=  1≤n≤u k r k (n), where u is a large real variable ( 1 ). For k = 2, the classic Gaussian circle problem, a detailed historical exposition can be found in the monograph of Kr¨ atzel [10]. The sharpest published results to date ( 2 ) read (1.1) R 2 (u)= πu 2 + P 2 (u), (1.2) P 2 (u)= O(u 46/73 (log u) 315/146 ), and ( 3 ) P 2 (u)= Ω − (u 1/2 (log u) 1/4 (log log u) (log 2)/4 (1.3) × exp(−c √ log log log u)) (c> 0), P 2 (u)= Ω + (u 1/2 exp(c ′ (log log u) 1/4 (log log log u) −3/4 )) (c ′ > 0). (1.4) 2000 Mathematics Subject Classification : 11P21, 11N37, 11L07. ( 1 ) Note that, in part of the relevant literature, t = u 2 is used as the basic variable. ( 2 ) Actually, M. Huxley has meanwhile improved further this upper bound, essentially replacing the exponent 46/73 = 0.6301 ... by 131/208 = 0.6298 ... The author is indebted to Professor Huxley for sending him a copy of his unpublished manuscript. ( 3 ) We recall that F 1 (u)= Ω * (F 2 (u)) means that lim sup u→∞ (*F 1 (u)/F 2 (u)) > 0 where * is either + or -, and F 2 (u) is positive for u sufficiently large. [191]