a1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (email: firstname.lastname@example.org)
a2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada (email: email@example.com)
a3 Instituto de Matemáticas, Universidad Nacional Autónoma de México, C. P. 58089, Morelia, Michoacán, México (email: firstname.lastname@example.org)
a4 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: email@example.com)
We give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+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≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where p is prime.
(Received December 23 2010)
(Accepted January 16 2012)
2010 Mathematics subject classification
Keywords and phrases
Friedlander was partially supported by NSERC Grant A5123 and Shparlinski was partially supported by ARC Grant DP1092835.
Communicated by F. Papparlardi and J. Shallit
Dedicated to the memory of Alf van der Poorten