Type: Article

Exponential and character sums with Mersenne numbers

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

Abstract:

We give new bounds on sums of the form Sigma(n <= N) Lambda(n) exp(2 pi iag(n)/m) and Sigma(n <= N) Lambda(n)chi(g(n) + a), where Lambda is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and chi is a multiplicative character modulo m. In particular, our results yield bounds on the sums Sigma(p <= N) exp(2 pi iaM(p)/m) and Sigma(p <= N) chi(M-p) with Mersenne numbers M-p = 2(p) - 1, where p is prime.