Revisited Carmichael’s Reduced Totient Function

: The modiﬁed Totient function of Carmichael λ ( . ) is revisited, where important properties have been highlighted. Particularly, an iterative scheme is given for calculating the λ ( . ) function. A comparison between the Euler ϕ and the reduced totient λ ( . ) functions aiming to quantify the reduction between is given.


Introduction
More than a century ago, Robert D. Carmichael (1879Carmichael ( -1967 [1] introduced a function λ(.) known as Carmichael's function. This λ(n) is spread as the reduced totient function which can be seen as the smallest divisor of Euler's totient function verifying Euler's theorem. This Totient function is deeply related to prime numbers and integer orders [2][3][4][5], mainly used for primality testing. Furthermore, the reader may cross in the literature that the Carmichael function represents the exponent (λ(n) represents the order of the largest cyclic subgroup of (Z/ n Z) * ) of the group (Z/ n Z) * .
In this paper, we aim to analyse λ(.) and we present some of its important properties. Mainly, we give in Lemma 3 a suitable iterative scheme for calculating the values of λ(n). In addition, we prove the following estimation λ(n) ≤ 1 2 N−1 ϕ(n), N is the number of odd prime divisors of n, which could be considered an indicator of reduction of the modified totient function and we can easily duduce that lim sup n→∞ ϕ(n) λ(n) = +∞.
The complexity of finding the inverse function of Carmichael λ(.) is more complex than finding the inverse of Euler function.

Preliminaries
In the literature, Carmichael's new totient function named λ is defined as follows: For the prime decomposition of a given natural integer otherwise.
We refer to [1,2,6,7] and references therein for the properties of the function λ. Figures 1 and 2 produce the first thousand values of λ(p) and ϕ(p). The points on the top lines represent λ(p) = p − 1 = ϕ(p), when p is a prime number.  In this section, we show how we built the modified Totient function λ. Euler theorem [1] states that if n and m are co-prime positive integers, then where ϕ(.) is Euler's Totient function. It is known that for any prime number p, we have ϕ(p) = p − 1 since all the positive integers less than p are co-prime with p. If p and q are two different primes, then The two integers k 1 and k 2 can be chosen in such a way that We can also conclude the following In the next definition, we introduce a new function related to the Totient function of Euler given as follows: According to the above definition, we will have: and by using previous results, we can conclude the following Lemma. • For all integer m, then Let us generalize the previous Lemma for n = Π N i=1 p i , ∀N ∈ N, but the function λ n = Π N i=1 p i needs also to be generalized. From Euler theorem, we can write Then, the function λ at n = Π N i=1 p i should be defined as follows: The following proposition provides a recursive scheme to evaluate the λ(n) for different situations.
Proposition 1. λ(n) can be calculated by a recursive way: . (5) and Again, we generalize our Lemma 1 result for n = p k i , where k ≥ 2, as follows: and Proof. The proof will be given after Lemma 5.
Proof. According to the definition of λ: and using the fact that and the smallest k 1 satisfying the above relation is the least common multiple (LCM) of , which completes the proof. where Proof. We will use the following two results: LCM(a 1 , · · · , a i+1 ) = LCM(LCM(a 1 , · · · , a i ), a i+1 ), and LCM(a, b) = a b gcd(x, y) .
According to Lemma 3 and (11), we have by setting Furthermore, according to (12), we obtain: which proves Lemma 4. we will obtain: . The Carmichael and Euler functions are a very important theoretic functions having a deep relationship with prime numbers. Figures 1 and 2 shows the first thousand values of λ(n) and ϕ(n), respectively, where the Euler function has been defined as ϕ(n) = (p n 1 1 − p k is the prime factorization of the integer n.

Properties of λ(.)
In this section, we present some properties of the new Totient function λ(.).

2.
If n = 2 Π k i=1 p k i i , p i are odd primes and k is any positive integer, then λ(n) = λ(n/2).
If p and q are two odd primes, and k and l are any natural numbers, then λ(p k q l ) ≤ 1 2 ϕ(p k q l ).

7.
If m, p, and q are three odd primes, and k, l and s are any natural numbers, then λ(m k p l q r ) ≤ 1 4 ϕ(m k p l q r ).
8. Let (p i ) i be an increasing sequence of primes, then:

1.
From the definition of the function λ(n), we can conclude the following:

2.
If n = 2Π k i=1 p k i i , p i are primes and k is any integer, then λ(n) = λ(n/2). Indeed,

4.
If n = 2 k , k > 2, then λ(2 k ) = ϕ(2 k )/2. We note the following Therefore, for any odd number m, we have If k > 3, we can factor more the term By the Euler theorem, we have m 2 k −2 k−1 − 1 ≡ 0 ( mod(2 5 ) ). By induction, we can easily prove that for any odd integer m and for all integer k > 2 the following statement is true:

5.
For n > 5, we have the following: which is even for any odd prime p. • λ(p k ) = p k−1 (p − 1), which is even for any odd prime p. 6. According to Lemma 3, we have: ≤ ϕ p k q l 2 , Property 1.

8.
Obvious, it is enough to prove that gcd λ ∏ k−1 i=1 p n i i , p k = 1.
Proof. From properties P6 and P7, the proof can be completed by induction. As a conclusion, we can easily prove the following limit lim sup n→∞ ϕ(n) λ(n) = +∞, by considering the subsequence n k = p 1 .p 2 . · · · .p k , where (p 1 , p 2 , . . . , p k ) are the first k-consecutive odd primes.
Again, since the primes are not bounded, we can conclude that lim inf n→∞ ϕ(n) λ(n) = 1.

Computations of λ(n) versus ϕ(n)
In this section, we compare the magnitude of the Euler function ϕ(n) versus the Carmichael function λ(n) see Tables 1 and 2. Figures 3 and 4 present, respectively, the ratio ϕ(n) λ(n) when n is a product of two primes, respectively, three primes.

Conclusions
In this paper, we presented how we built the modified Totient function of Carmichael λ(.). Important properties have been highlighted, particularly the given iterative scheme for calculating the λ(.) function. Some preliminary numerical results comparing the Euler ϕ and the reduced totient λ(.) functions aiming to quantify the reduction between them are given (see Tables 1 and 2  {n : λ(n) = k}, k is a given positive integer.
Furthermore, it may be worthwhile investigating more results in Corollary 1 by finding a better upper-bound.