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On the distribution of angles of the Salié sums

Published online by Cambridge University Press:  17 April 2009

Igor E. Shparlinski
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
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For a prime p and integers a and b, we consider Salié sums

where χ2(x) is a quadratic character and x¯ is the modular inversion of x, that is, xx¯≡ 1 (mod p). One can naturally associate with Sp (a, b) a certain angle θp(a, b) ∈ [0, π]. We show that, for any fixed ε > 0, these angles are uniformly distributed in [0, π] when a and b run over arbitrary sets , ℬ ⊆ {0, 1, …, p − 1} such that there are at least p1+ε quadratic residues modulo p among the products ab, where (a, b) ∈  × ℬ.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 2 , April 2007 , pp. 221 - 227
Copyright
Copyright © Australian Mathematical Society 2007

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