Abstract
We develop a substantial enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree k for moments of order 2s, thereby obtaining for the first time near-optimal estimates for s>(5/8)k^2. There are numerous applications. In particular, when k is large, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s⩾1.543k^2.
Original language | English |
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Pages (from-to) | 519-560 |
Number of pages | 42 |
Journal | Proceedings of the London Mathematical Society |
Volume | 111 |
Issue number | 3 |
Early online date | 15 Jul 2015 |
DOIs | |
Publication status | Published - 3 Sep 2015 |
Keywords
- math.NT
- 11L15
- 11L07
- 11P05
- 11P55