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On similar short sums

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  26 February 2010

N. Watt
N. Watt
Affiliation:
45 Charles Way, Limekilns, Fife. KY11 3LH, United Kingdom
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Extract

In the paper [5] bounds are found for the sum,

for a suitable Dirichlet character χ mod r, and real functionf(x). The proofs in that paper use Bombieri and Iwaniec's method [1], one formulation of which has as part of its first step the estimation of S in terms of a sum of many shorter sums of the form,

where e(x) = exp (2πix), mi∈ [M, 2M], and each mi, lies in its own interval, of length N ≥ M/4, that is disjoint from those of the others. This paper addresses a problem springing from above: to bound the numbers of ‘similar’ pairs, Si+, Si+, satisfying both

and

where ‖x‖ = min{|x − n|: n ∈ ℤ}. Lemma 5.2.1 of [3] (partial summation) shows that each sum in a similar pair is a bounded multiple of the other.

MSC classification

Secondary: 11L40: Estimates on character sums
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 193 - 204
Copyright
Copyright © University College London 1999

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References

1.Bombieri, E. and Iwaniec, H.. On the order of ζ(½†it). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 449–472.Google Scholar
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5.Huxley, M. N. and Watt, N. Hybrid bounds for Dirichlet's L-function. To appear in Math. Proc. Cam. Phil. Soc..Google Scholar
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