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2023•Bulletin of the Australian Mathematical SocietyOpen access

ON SUMS INVOLVING THE EULER TOTIENT FUNCTION

Isao Kiuchi, YUKI TSURUTA

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Abstract

Abstract Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $

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Abstract Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $

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Available abstract

Abstract Let $\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers $n_{1},\ldots ,n_{k}$ and let $\phi $ be the Euler totient function. For any real number $x>3$ and any integer $k\geq 2$ , we investigate the asymptotic behaviour of $\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $

Key concepts: Euler's totient function, Mathematics, Combinatorics, Euler's formula, Integer (computer science), Greatest common divisor, Function (biology), Arithmetic function

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