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The asymptotic formula in Waring's problem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Kent D. Boklan
Kent D. Boklan
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
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Extract

Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation

in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula

whenever s≥(k−2)2k−1 + 5. Here , the singular series, relates the local solubility of (1.1). For each k we define to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.

MSC classification

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 329 - 347
Copyright
Copyright © University College London 1994

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References

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