Journal of the Australian Mathematical Society

Table of Contents - February 2012 - Volume 92, Issue 01  

Research Article

EXPONENTIAL AND CHARACTER SUMS WITH MERSENNE NUMBERS

WILLIAM D. BANKSa1, JOHN B. FRIEDLANDERa2, MOUBARIZ Z. GARAEVa3 and IGOR E. SHPARLINSKIa4 c1

a1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (email: bankswd@missouri.edu)

a2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada (email: frdlndr@math.toronto.edu)

a3 Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 58089, Morelia, Michoacán, México (email: garaev@matmor.unam.mx)

a4 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)

Abstract

We give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+a), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums ∑ p≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where p is prime.

(Received December 23 2010)

(Accepted January 16 2012)

2010 Mathematics subject classification

  • primary 11L07; secondary 11L20

Keywords and phrases

  • Mersenne number;
  • exponential sums;
  • character sums

Correspondence:

c1 For correspondence; e-mail: igor.shparlinski@mq.edu.au

Footnotes

Friedlander was partially supported by NSERC Grant A5123 and Shparlinski was partially supported by ARC Grant DP1092835.

Communicated by F. Papparlardi and J. Shallit

Dedicated to the memory of Alf van der Poorten