On sums of two cubes: an Ω₊-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 kth 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 $P_{_k}(t)$ which is known to be $O(t^{1/4})$ in mean-square. In this article it is proved that $P_{₃}(t) = Ω₊(t^{1/4}(loglog t)^{1/4})$ as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently large subset which is linearly independent over ℚ

Pseudorandomness of a Random Kronecker Sequence

International audienceWe study two randomness measures for the celebrated Kroneckersequence S(!) formed by the fractional parts of the multiples ofa real !. The first measure is the well-known discrepancy, whereas theother one, the Arnold measure, is less popular. Both describe the behaviourof the truncated sequence ST (!) formed with the first T terms,for T !". We perform a probabilistic study of the pseudorandomnessof the sequence S(!) (discrepancy and Arnold measure), and we giveestimates of their mean values in two probabilistic settings : the input !may be either a random real or a random rational. The results exhibitstrong similarities between the real and rational cases; they also showthe influence of the number T of truncated terms, via its relation to thecontinued fraction expansion of !