Exponential sums over Mersenne numbers

Abstract

We give estimates for exponential sums of the form P n6N �( n)exp(2 �iag n /m ), where m is a positive integer, a and g are integers relatively prime to m, andis the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form P p6N exp(2 �iaM p/m ), where Mp is the Mersenne number; Mp = 2 p !1 for any prime p. We also estimate some closely related sums, including P n6N µ(n)exp(2 �iag n /m ) and P n6N µ 2 (n)exp(2 �iag n /m ), where µ is the Mobius function.