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The Quarterly Journal of Mathematics Advance Access published online on July 11, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap023
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

EXPONENTIAL SUMS WITH CONSECUTIVE MODULAR ROOTS OF AN INTEGER

Igor E. Shparlinski{dagger}

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

{dagger} E-mail: igor{at}ics.mq.edu.au

Received 8 January 2009; revised 11 June 2009
  Abstract

J. Bourgain and the author have recently estimated exponential sums with consecutive modular roots {vartheta}1/n (mod p), where {vartheta} is of multiplicative order t ≥ p{varepsilon} modulo a prime p (for some fixed {varepsilon} > 0) and n runs through the integers in the interval [M + 1, M + N] with gcd(n, t) = 1. However, the saving in that bound against the trivial estimate has not been made explicit. It is shown here that for t ≥ p1/2+{varepsilon} one can obtain a fully explicit bound for such exponential sums.


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