Some hypotheses in Waring’s problem. (English) Zbl 0608.10044
Number theory and combinatorics, Proc. Conf., Tokyo/Jap., Okayama/Jap. and Kyoto/Jap. 1984, 105-117 (1985).
[For the entire collection see Zbl 0601.00003.]
The author considers the usual minor arcs in the treatment of Waring’s problem by the circle method. A hypothesis is introduced which in its most powerful form states \[ \sum_{1\leq m\leq M}m^{-}\quad \exp (- i\phi m^{k/(k-1)})\ll \phi^{(k-1)/(2k)} M^{\epsilon} \] as \(M\to \infty\), for each \(\epsilon >0\). This would imply that each large enough N is expressible as a sum of \(2k+1\) kth powers of positive integers, provided the necessary congruence conditions are satisfied so that the usual singular series is positive.
The author considers the usual minor arcs in the treatment of Waring’s problem by the circle method. A hypothesis is introduced which in its most powerful form states \[ \sum_{1\leq m\leq M}m^{-}\quad \exp (- i\phi m^{k/(k-1)})\ll \phi^{(k-1)/(2k)} M^{\epsilon} \] as \(M\to \infty\), for each \(\epsilon >0\). This would imply that each large enough N is expressible as a sum of \(2k+1\) kth powers of positive integers, provided the necessary congruence conditions are satisfied so that the usual singular series is positive.
Reviewer: G.Greaves
