Type: Article
Publication Date: 2012-02-01
Citations: 9
DOI: https://doi.org/10.1017/s1446788712000109
Abstract We give new bounds on sums of the form ∑ n ≤ N Λ( n )exp (2 πiag n / m ) and ∑ n ≤ N Λ( n ) χ ( g n + a ), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m , and χ is a multiplicative character modulo m . In particular, our results yield bounds on the sums ∑ p ≤ N exp (2 πiaM p / m ) and ∑ p ≤ N χ ( M p ) with Mersenne numbers M p =2 p −1, where p is prime.