Exponential and character sums with mersenne numbers

William D. Banks; John B. Friedlander; Moubariz Z. Garaev; Igor E. Shparlinski

doi:10.1017/S1446788712000109

Exponential and character sums with mersenne numbers

William D. Banks, John B. Friedlander, Moubariz Z. Garaev, Igor E. Shparlinski^*

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

28 Downloads (Pure)

Abstract

We give new bounds on sums of the form Σ _n≤N ∧ (n) exp (2πiag ⁿ/m) and Σ _n≤N ∧ (n)Χ(g ⁿ + a), where ∧ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and Χ is a multiplicative character modulom. In particular, our results yield bounds on the sums Σ _p≤N exp (2πiaM _p/m) and Σ _p≤N Χ(M _p) with Mersenne numbers M _p = 2 ^p - 1, where p is prime.

Original language	English
Pages (from-to)	1-13
Number of pages	13
Journal	Journal of the Australian Mathematical Society
Volume	92
Issue number	1
DOIs	https://doi.org/10.1017/S1446788712000109
Publication status	Published - Feb 2012

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title = "Exponential and character sums with mersenne numbers",

abstract = "We give new bounds on sums of the form Σ n≤N ∧ (n) exp (2πiag n/m) and Σ n≤N ∧ (n)Χ(g n + a), where ∧ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and Χ is a multiplicative character modulom. In particular, our results yield bounds on the sums Σ p≤N exp (2πiaM p/m) and Σ p≤N Χ(M p) with Mersenne numbers M p = 2 p - 1, where p is prime.",

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AU - Friedlander, John B.

AU - Garaev, Moubariz Z.

AU - Shparlinski, Igor E.

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N2 - We give new bounds on sums of the form Σ n≤N ∧ (n) exp (2πiag n/m) and Σ n≤N ∧ (n)Χ(g n + a), where ∧ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and Χ is a multiplicative character modulom. In particular, our results yield bounds on the sums Σ p≤N exp (2πiaM p/m) and Σ p≤N Χ(M p) with Mersenne numbers M p = 2 p - 1, where p is prime.

AB - We give new bounds on sums of the form Σ n≤N ∧ (n) exp (2πiag n/m) and Σ n≤N ∧ (n)Χ(g n + a), where ∧ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and Χ is a multiplicative character modulom. In particular, our results yield bounds on the sums Σ p≤N exp (2πiaM p/m) and Σ p≤N Χ(M p) with Mersenne numbers M p = 2 p - 1, where p is prime.

UR - https://www.scopus.com/pages/publications/84862532380

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JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

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ER -

Exponential and character sums with mersenne numbers

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