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A SIEVE PROBLEM AND ITS APPLICATION

Part of: Multiplicative number theory

Published online by Cambridge University Press:  27 September 2016

Andreas Weingartner
Andreas Weingartner*
Affiliation:
Department of Mathematics, Southern Utah University, 351 West University Boulevard, Cedar City, UT 84720, U.S.A. email weingartner@suu.edu
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Abstract

Let $\unicode[STIX]{x1D703}$ be an arithmetic function and let ${\mathcal{B}}$ be the set of positive integers $n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{k}^{\unicode[STIX]{x1D6FC}_{k}}$ which satisfy $p_{j+1}\leqslant \unicode[STIX]{x1D703}(p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{j}^{\unicode[STIX]{x1D6FC}_{j}})$ for $0\leqslant j<k$. We show that ${\mathcal{B}}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of ${\mathcal{B}}$. When $\unicode[STIX]{x1D703}(n)/n$ is non-decreasing, the set ${\mathcal{B}}$ coincides with the set of integers $n$ whose divisors $1=d_{1}<d_{2}<\cdots <d_{\unicode[STIX]{x1D70F}(n)}=n$ satisfy $d_{j+1}\leqslant \unicode[STIX]{x1D703}(d_{j})$ for $1\leqslant j<\unicode[STIX]{x1D70F}(n)$.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 213 - 229
Copyright
Copyright © University College London 2016 

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