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The average orders of Hooley's Δr-functions
Published online by Cambridge University Press: 26 February 2010
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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.
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