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Bilinear sums with exponential functions

Author(s): Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 137 (2009), 2217-2224.
MSC (2000): Primary 11L07, 11L26
Posted: March 4, 2009
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Abstract: Let $ g\ne 0, \pm 1$ be a fixed integer. Given two sequences of complex numbers $ \left(\varphi_m\right)_{m=1}^\infty$ and $ \left(\psi_n\right)_{n=1}^\infty $ and two sufficiently large integers $ M$ and $ N$, we estimate the exponential sums

$\displaystyle \sum_{\substack{p \le M \mathrm{gcd}(ag,p) =1}} \sum_{1 \le n \le N} \varphi_p\psi_n \mathbf{e}_p\left(a g^n\right), \qquad a \in \mathbb{Z}, $

where the outer summation is taken over all primes $ p \le M$ with $ \mathrm{gcd}(ag,p) =1$.

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Additional Information:

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor@ics.mq.edu.au

DOI: 10.1090/S0002-9939-09-09882-7
PII: S 0002-9939(09)09882-7
Received by editor(s): September 17, 2008
Posted: March 4, 2009
Additional Notes: During the preparation of this paper, the author was supported in part by ARC grant No. DP0556431.
Communicated by: Wen-Ching Winnie Li
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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