Exponential sums over Mersenne numbers

Abstract

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