Functiones et Approximatio Commentarii Mathematici

Arithmetic functions and their coprimality

Jean-Marie De Koninck and Imre Kátai

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Abstract

Let $D\ge 3$ be an odd integer and $\ell\ge -1$ be a non zero integer such that gcd$(\ell,D)=1$. Let $f,g:\mathbb{N} \to \mathbb{N}$ be multiplicative functions such that $f(p)=D$ and $g(p)=p+\ell$ for each prime $p$. We estimate the number of positive integers $n\le x$ such that gcd$(f(n),g(n))=1$. If $D$ is a prime larger than 3, we also examine the size of the number of positive integers $n\le x$ for which $\mbox{gcd}(g(n),f(n-1))=1$.

Article information

Source
Funct. Approx. Comment. Math., Volume 45, Number 1 (2011), 55-66.

Dates
First available in Project Euclid: 26 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1317045231

Digital Object Identifier
doi:10.7169/facm/1317045231

Mathematical Reviews number (MathSciNet)
MR2865412

Zentralblatt MATH identifier
1261.11062

Subjects
Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11N37: Asymptotic results on arithmetic functions

Keywords
arithmetic functions number of divisors sum of divisors shifted primes

Citation

De Koninck, Jean-Marie; Kátai, Imre. Arithmetic functions and their coprimality. Funct. Approx. Comment. Math. 45 (2011), no. 1, 55--66. doi:10.7169/facm/1317045231. https://projecteuclid.org/euclid.facm/1317045231


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References

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