Exponential sums over Mersenne numbers

We give estimates for exponential sums of the form Sigma(nless than or equal toN) Lambda(n) exp(2piiag(n)/m), where m is a positive integer, a and g are integers relatively prime to m, and Lambda is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form Sigma(pless than or equal toN) exp(2piiaM(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 Sigma(nless than or equal toN) mu(n) exp(2piiag(n)/m) and Sigma(nless than or equal toN) mu(2)(n) exp(2piiag(n)/m), where mu is the Mobius function.