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A special triple exponential sum

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  26 February 2010

H.-Q. Liu
H.-Q. Liu
Affiliation:
206-10, Bao Guo St., Harbin, 150066, China.
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Extract

In this short paper, we shall give a new estimate for the exponential sum S(H, M, N), where

e( ξ,) = exp (2πiξ;) for a real number ξ, am and bn are complex numbers with |am| ≤ 1 and |bn| ≤ l, H, M, N ≤1, , x is a large number, ε is a sufficiently small positive number, and Y ≤ x(½)−ε (h ∼ H means 1≤h/H < 2 and so on). In making application of the Rosser-Iwaniec linear sieve of Iwaniec [6] to find almost primes in short intervals of the type (x − y, x], Halberstam, Heath-Brown and Richert [4] first considered an estimate for S(H, M, N) to the effect that

with MN as large as possible. Later, better estimates were given in Iwaniec and Laborde [7], and Fouvry and Iwaniec [3]. Of course, the most interesting case would be finding P2 numbers in a short interval (x − y, x]. The related estimate of [7] implies that (1) holds provided that

MSC classification

Secondary: 11L07: Estimates on exponential sums
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 137 - 143
Copyright
Copyright © University College London 1995

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References

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