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APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV’S MEAN VALUE THEOREM

Part of: Exponential sums and character sums Additive number theory; partitions

Published online by Cambridge University Press:  21 December 2016

Trevor D. Wooley
Trevor D. Wooley*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, U.K. email matdw@bristol.ac.uk
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Abstract

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$.

MSC classification

Primary: 11L15: Weyl sums 11L07: Estimates on exponential sums 11P55: Applications of the Hardy-Littlewood method
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 292 - 350
Copyright
Copyright © University College London 2016 

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