Vinogradov's mean value theorem via efficient congruencing, II

Details Vinogradov's mean value theorem via efficient congruencing, II Vinogradov's mean value theorem via efficient congruencing, II

2011-12-01

arxiv.org/abs/1112.0358v1

Mathematics

Number Theory

Scientific paper

We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1<=s<=k^2-1. Thereby, we show that quasi-diagonal behaviour holds when s=o(k^2), we obtain near-optimal estimates for 1<=s<=(1/4)k^2+k, and optimal estimates for s>=k^2-1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s>=2k^2-2k-8 (k>=6).

Affiliated with

Also associated with

No associations

Rating

Vinogradov's mean value theorem via efficient congruencing, II does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Vinogradov's mean value theorem via efficient congruencing, II, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Vinogradov's mean value theorem via efficient congruencing, II will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-139569