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Yet Another Generalization of
Euler’s Totient Function
Matthew Holden, Michael Orrison, Michael Vrable
For every positive integer n, Euler’s totient function, or φ-function, gives the number φ(n) of integers less than n that are relatively prime to n, with the convention that φ(1) = 1. Students of abstract algebra also know φ(n) as the number of generators of the cyclic group Z/nZ. It therefore seems worthwhile to consider generalizations of Euler’s totient function from a group theoretic perspective. One such generalization is Jordan’s totient function [2, pp. 147-155]. For pos- itive integers n and k, Jk(n) is defined to be the number of k-tuples (a 1 ,... , ak) from { 1 ,... , n} such that the greatest common divisor of {a 1 ,... , ak} is rela- tively prime to n. Note that Jk is a generalization Euler’s totient function since J 1 (n) = φ(n). To view Jordan’s totient function from a group theoretic perspective, note that Jk(n) also counts the number of sequences (g 1 ,... , gk) of elements in Z/nZ such that, if Gi is the subgroup generated by {g 1 ,... , gi}, then
{ 0 } ≤ G 1 ≤ · · · ≤ Gk− 1 ≤ Gk = Z/nZ.
Moreover, by using simple properties of subgroups and quotient groups of Z/nZ, identities concerning Jk(n) may be obtained. For example, Gegenbauer (see [2, p. 151]) showed that
Jk+l(n) =
1 ≤d≤n d|n
dlJk(d)Jl(n/d).
To see why Gegenbauer’s result is true, recall that for each divisor d of n, there is a unique subgroup of order d in Z/nZ. Furthermore, the corresponding quotient group (Z/nZ)/(Z/dZ) is isomorphic to Z/(n/d)Z. There are Jk(d) sequences of length k for Z/dZ. Every extension of such a sequence to a sequence of length k + l for Z/nZ corresponds to a sequence of length l for the quotient group. Since every element of the quotient group has d representatives in Z/nZ, the number of sequences of length k+l that pass through Z/dZ is dlJk(d)Jl(n/d). Summing over all divisors gives Gegenbauer’s result. (See [2] or [4] for other identities involving Jk(n).) In this note, we consider a variation of Jordan’s totient function defined as follows. For positive integers n and k, let Mk(n) be the number of sequences
(g 1 ,... , gk) of elements in Z/nZ such that, if Gi is the subgroup generated by {g 1 ,... , gi}, then
{ 0 } < G 1 < · · · < Gk− 1 < Gk = Z/nZ.
In other words, Mk(n) counts only those sequences (g 1 ,... , gk) with the property that Gi is strictly contained in Gi+1 for i = 1,... , k − 1. Together with the convention that M 1 (1) = 1, we have that M 1 (n) = φ(n). The function Mk(n) is therefore another generalization of Euler’s totient function. One noteworthy feature of Mk(n) is that, for a fixed n, Mk(n) will eventually become 0. In fact, if n = pe 11 · · · pe rr where the pi are prime, then Mk(n) = 0 for all k > e 1 + · · · + er. This of course follows from the fact that there are no appropriate sequences of subgroups of length more than e 1 + · · · + er. Unlike Jk, however, Mk is not multiplicative, i.e., Jk(m)Jk(n) need not equal Jk(mn) when m and n are relatively prime. For example, M 2 (6) = 10 while M 2 (2) = M 2 (3) = 0. We conclude this note with some additional properties of Mk.
Theorem 1. If n, k and l are positive integers, then
Mk+l(n) =
1 <d<n d|n
dlMk(d)Ml(n/d).
Proof. The argument is similar to that for Gegenbauer’s result, the only change being that the sum is now over the nontrivial divisors of n due to the strict containment of the corresponding subgroups.
Corollary 2. If n and k are positve integers, and p is prime, then
Mk+1(pn) = (p − 1)pn−^1
n∑− 1
j=k
Mk(pj^ ).
Proof. By Theorem 1,
Mk+1(pn) =
1 <d<pn d|pn
dMk(d)M 1 (n/d)
n∑− 1
j=k
pj^ Mk(pj^ )φ(pn−j^ )
n∑− 1
j=k
(p − 1)pn−^1 Mk(pj^ )
= (p − 1)pn−^1
n∑− 1
j=k
Mk(pj^ ).
than n. Define L(pn) =
∑n k=1 Mk(p
n). By Corollary 2 we have
∑^ n
k=
Mk(pn) = M 1 (pn) +
∑^ n
k=
Mk(pn)
= pn−^1 (p − 1) +
∑^ n
k=
pn−^1 (p − 1)
n∑− 1
j=k− 1
Mk− 1 (pj^ )
= pn−^1 (p − 1) + pn−^1 (p − 1)
n∑− 1
i=
L(pi)
= pn−^1 (p − 1)
n∑− 1
i=
L(pi)
By induction we have
n∑− 1
i=
L(pi) =
n∑− 2
i=
L(pi)
n∏− 1
k=
pk−^2 (p − 1) + 1
n∏− 1
k=
pk−^2 (p − 1) + 1
1 + pn−^2 (p − 1)
) n∏−^1
k=
pk−^2 (p − 1) + 1
∏^ n
k=
pk−^2 (p − 1) + 1
The theorem follows immediately.
References
[1] L. Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht,
[2] L. Dickson, History of the theory of numbers. Vol. I: Divisibility and pri- mality, Chelsea Publishing Co., New York, 1966.
[3] P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134–151.
[4] R. Sivaramakrishnan, The many facets of Euler’s totient. II. Generaliza- tions and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169–187.
