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On similar short sums
Part of:
Exponential sums and character sums
Published online by Cambridge University Press: 26 February 2010
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
- Copyright
- Copyright © University College London 1999
References
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