On Waring's problem: some consequences of Golubeva's method

Abstract

We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the Hardy-Littlewood method.2010 Mathematics Subject Classification. 11P05, 11D85.

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

“…Proof of Theorem 3. We keep the notation from Wooley [19,Section 3] and modify only the parts concerning the bound of Golubeva's theorem. The necessary requirements to apply Theorem 2, piq N M 12 ą p 17 , piiq N M 6 ą p 11 and piiiq N ą p 3 , are fulfilled provided that (cf.…”

“…We singled out the case of quadratic χ because this is the relevant case for quadratic forms and the main application that we proceed to present. It has been investigated by Wooley [19] under which conditions on the exponents k j , j " 1, . .…”

“…, k t are not all even. The original bounds in[19, Theorem 1.2] are γpkq ă 12{17 with the assumption of the Riemann hypothesis,γpkq ă 74{105 for piq and γpkq ă 74{105 for piiq. As a consequence, it follows that every sufficiently large number n is represented in the form if the truth of the Riemann hypothesis is assumed.Acknowledgments.…”

Exceptional Sets in Waring’s Problem: Two Squares, Two Cubes and Two Sixth Powers

“…T. D. Wooley [7] applied Golubeva's method to show, subject to the truth of the generalized Riemann hypothesis, that R(n) > 0 for all large integers n. However, his method fails to obtain the anticipated asymptotic formula for R(n).…”

On the Asymptotic Formula in Waring's Problem: One Square and Three Fifth Powers

“…For a thorougher account of Waring's problem with unlike powers the interested reader may consult the work of Hooley [8], chapter 8 of Vaughan [16] and the extensive references therein, and the recent papers of Wooley [18] and Friedlander and Wooley [5].…”