Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.16

Exponential Sums Involving the k-th Largest Prime Factor Function


Jean-Marie De Koninck
Département de Mathématiques
Université Laval
Québec G1V 0A6
Canada

Imre Kátai
Computer Algebra Department
Eötvös Loránd University
1117 Budapest
Pázmány Péter Sétány I/C
Hungary

Abstract:

Letting Pk(n) stand for the k-th largest prime factor of $n\ge 2$ and given an irrational number $\alpha$ and a multiplicative function f such that |f(n)|=1 for all positive integers n, we prove that $\sum_{n\le x} f(n) \exp\{2\pi i \alpha P_k(n)\}=o(x)$ as $x\to \infty$.

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Received October 7 2012; revised version received October 27 2012. Published in Journal of Integer Sequences, March 2 2013.

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