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The average orders of Hooley's Δr-functions

Published online by Cambridge University Press:  26 February 2010

R. R. Hall  and
G. Tenenbaum
R. R. Hall
Affiliation:
Department of Mathematics, University of York, Heslington, York. YO1 5DD
G. Tenenbaum
Affiliation:
UER Sciences Mathématiques, Université de Nancy 1, Boîte Postale 239, 54506 Vandoeuvre les Nancy Cedex, France.
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Extract

In this paper we are concerned with upper bounds for the sums

where

and Δ2(n) is written simply as Δ(n). These functions were introduced by Hooley [4] and applied in a novel way to problems related to Waring's, and in Diophantine approximation. Thus Hooley deduced from his result about S2(x) that for any irrational θ, real γ, and fixed ε > 0, the inequality

holds for infinitely many n. His result for S3(x) led to a proof that

where r8(n) denotes the number of representations of n as the sum of eight positive cubes.

Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 98 - 109
Copyright
Copyright © University College London 1984

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References

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