Sums over Numbers with Restricted Prime Factors

Walid Wannes,Khadija Mbarki

doi:10.12691/tjant-5-5-2

Abstract

Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

Highlights

0 ≤ b ≤ q j ) and is the set of non negative integers. Such functions were introduced by Gelfond [7] and further studied by Coquet [1], Kátai [9]
S( p)≡a mod b and Ω (n) = Ω a,b,q (n) = ∑ 1. pα |n S( p)≡a mod b. Both functions have been studied in [12,13], it was proved that their normal order is 1 log log n
The aim of this paper is to provide asymptotic formulas for sums involving ω (n) and Ω (n)

Summary

Introduction

We discuss sums of the form ∑ f (n)ω (n), where f is a n≤x multiplicative function. The sum of digits function in basis q is defined by Both functions have been studied in [12,13], it was proved that their normal order is 1 log log n.

Objectives

Results

Conclusion