Menon-type identities again: a note on a paper by Li, Kim and Qiao. (English) Zbl 1463.11003
Here is the authors’ abstract: “We give common generalizations of the Menon-type identities by R. Sivaramakrishnan [J. Indian Math. Soc., New Ser. 33, 127–132 (1969; Zbl 0206.33406)] and Y. Li et al. [Publ. Math. 94, No. 3–4, 467–475 (2019; Zbl 1438.11003)]. Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character-free approach for the proof.”
For example, the authors present a formula for \[\sum_{1\le a_1\le n}\cdots \sum_{1\le a_k\le n}[(a_1-1,n),\dots, (a_k-1,n)],\] which for \(k=1\) yields Menon’s classical identity: \[M(n):=\sum_{a=1}^n (a-1, n) =\varphi(n) \tau(n). \] Here \(n\) and \(k\) are positive integers, \((\ ,\,)\) denotes the greatest common divisor, \([\ , \ ]\) denotes the least common multiple, \(\varphi\) is Euler’s totient function, and \(\tau(n)\) is the sum of the divisors of \(n\).
For example, the authors present a formula for \[\sum_{1\le a_1\le n}\cdots \sum_{1\le a_k\le n}[(a_1-1,n),\dots, (a_k-1,n)],\] which for \(k=1\) yields Menon’s classical identity: \[M(n):=\sum_{a=1}^n (a-1, n) =\varphi(n) \tau(n). \] Here \(n\) and \(k\) are positive integers, \((\ ,\,)\) denotes the greatest common divisor, \([\ , \ ]\) denotes the least common multiple, \(\varphi\) is Euler’s totient function, and \(\tau(n)\) is the sum of the divisors of \(n\).
Reviewer: Moshe Roitman (Haifa)
MSC:
11A07 | Congruences; primitive roots; residue systems |
11A25 | Arithmetic functions; related numbers; inversion formulas |
Keywords:
Menon’s identity; gcd-sum function; arithmetic function of several variables; multiplicative function; lcm function; polynomial with integer coefficientsOnline Encyclopedia of Integer Sequences:
Pillai’s arithmetical function: Sum_{k=1..n} gcd(k, n).a(n) = d(n) * phi(n), where d(n) is the number of divisors function.