Bulletin of the Australian Mathematical Society

Table of Contents - June 2011 - Volume 83, Issue 03  

Research Article

BOUNDS OF MULTIPLICATIVE CHARACTER SUMS WITH FERMAT QUOTIENTS OF PRIMES

IGOR E. SHPARLINSKIa1

a1 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)

Abstract

Given a prime p, the Fermat quotient qp(u) of u with gcd (u,p)=1 is defined by the conditions

\[ q_p(u) \equiv \frac {u^{p-1} -1}{p}\mod p, \quad 0 \le q_p(u) \le p-1. \]

We derive a new bound on multiplicative character sums with Fermat quotients qp(ℓ) at prime arguments ℓ.

(Received August 14 2010)

(Online publication February 07 2011)

2010 Mathematics subject classification

  • primary 11A07; secondary 11L40;
  • 11N25

Keywords and phrases

  • Fermat quotients;
  • character sums;
  • Vaughan identity

Footnotes

The author was supported in part by Australian Research Council Grant DP1092835.