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EXPONENTIAL AND CHARACTER SUMS WITH MERSENNE NUMBERS

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  26 April 2012

WILLIAM D. BANKS ,
JOHN B. FRIEDLANDER ,
MOUBARIZ Z. GARAEV  and
IGOR E. SHPARLINSKI
WILLIAM D. BANKS
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (email: bankswd@missouri.edu)
JOHN B. FRIEDLANDER
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada (email: frdlndr@math.toronto.edu)
MOUBARIZ Z. GARAEV
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 58089, Morelia, Michoacán, México (email: garaev@matmor.unam.mx)
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)
*
✉For correspondence; e-mail: igor.shparlinski@mq.edu.au
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Abstract

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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.

Keywords

Mersenne numberexponential sumscharacter sums

MSC classification

Secondary: 11L07: Estimates on exponential sums
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 1 - 13
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

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

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