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arxiv: 1101.0574 · v1 · pith:WCHRH2HPnew · submitted 2011-01-03 · 🧮 math.NT

Vinogradov's mean value theorem via efficient congruencing

classification 🧮 math.NT
keywords conjecturedvinogradovapplicationsapproachasymptoticbestboundscongruencing
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We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's problem holds for sums of s kth powers of natural numbers whenever s is at least 2k^2+2k-3.

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