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The Quarterly Journal of Mathematics Advance Access originally published online on August 2, 2008
The Quarterly Journal of Mathematics 2009 60(4):401-411; doi:10.1093/qmath/han023
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS

William D. Banks {dagger}

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Igor E. Shparlinski{ddagger}

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

{ddagger} Corresponding author. E-mail: igor{at}ics.mq.edu.au

Received 7 March 2008;
  Abstract

For a nontrivial multiplicative character {chi} modulo p, we bound character sums


Formula

taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f(x) = x{eta} with some positive real number {eta}, our bounds extend recent results of several authors.

{dagger} E-mail: bbanks{at}math.missouri.edu


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