Journal de Th´eoriedes Nombres de Bordeaux 16 (2004), 107–123 Average order in cyclic groups par Joachim von zur GATHEN, Arnold KNOPFMACHER, Florian LUCA, Lutz G. LUCHT et Igor E. SHPARLINSKI Resum´ e.´ Pour chaque entier naturel n, nous d´eterminons l’ordre moyen α(n) des ´el´ements du groupe cyclique d’ordre n. Nous mon- trons que plus de la moiti´ede la contribution `a α(n) provient des ϕ(n) ´el´ements primitifs d’ordre n. Il est par cons´equent int´eressant d’´etudier ´egalement la fonction β(n) = α(n)/ϕ(n). Nous d´etermi- nons le comportement moyen de α, β, 1/β et consid´erons aussi ces fonctions dans le cas du groupe multiplicatif d’un corps fini. Abstract. For each natural number n we determine the average order α(n) of the elements in a cyclic group of order n. We show that more than half of the contribution to α(n) comes from the ϕ(n) primitive elements of order n. It is therefore of interest to study also the function β(n) = α(n)/ϕ(n). We determine the mean behavior of α, β, 1/β, and also consider these functions in the multiplicative groups of finite fields. Section 1. Introduction For a positive integer n, we determine the average order α(n) of the elements in the additive cyclic group Zn of order n. The major contribution to α(n) is from the ϕ(n) primitive elements in Zn, each of order n. We show that, in fact, the other elements never contribute more than the primitive ones do. More precisely, we consider the relative version β(n) = α(n)/ϕ(n). With ζ(2)ζ(3) 315 ζ(3) A = = ≈ 1.94359 64368 ζ(6) 2π4 we have for n ≥ 2: 1 = lim inf β(n) < β(n) < lim sup β(n) = A. n−→∞ n−→∞ We also determine the mean behavior of α, β, and 1/β, and discuss the average order of elements in the multiplicative groups of finite fields. The lower bounds for β are different for even and for odd characteristic. Manuscrit re¸cule 19 avril 2002. 108 J. von zur Gathen, A. Knopfmacher, F. Luca, L. Lucht, I. Shparlinski The original motivation for this research was the usage of groups in cryptography. Here one looks for cyclic groups of large order (preferably a prime number). If we take a finite field and pick a random element from it, how large can we expect its order to be? Intuition says that one should avoid fields whose multiplicative group order is largely made up from small prime factors. The results of this paper put this intuition on a firm basis. Section 2. The average order For a ∈ Zn, we denote by ord(a) its order in the additive group Zn. Then ord(a) divides n, and for each divisor d of n, there are exactly ϕ(d) elements in Zn of order d. Thus the average order in Zn is 1 X 1 X α(n) = ord(a) = dϕ(d). n n a∈Zn d|n The main contribution is the term with d = n, and we normalize by it: α(n) β(n) = . ϕ(n) Since 1/n and ϕ(n) are multiplicative functions of n, so is their Dirichlet convolution α(n) (see Apostol 1976, Theorem 2.14), and also β(n). We first determine their values in the case of a prime power. Lemma 2.1. Let p be a prime and k ≥ 1 an integer. Then pk+1 1 1 1 α(pk) = + , β(pk) = 1 + 1 + . p + 1 pk(p + 1) p2 − 1 p2k−1 In particular β(1) = 1 < β(pk+1) < β(pk) ≤ β(p) = 1 + 1/(p2 − p). Proof. We have 1 X 1 X α(pk) = piϕ(pi) = 1 + (p − 1) · p2i−1 pk pk 0≤i≤k 1≤i≤k p2k+1 + 1 = , pk(p + 1) p2k+1 + 1 1 1 β(pk) = = 1 + 1 + . pk(p + 1)(p − 1)pk−1 p2 − 1 p2k−1 Theorem 2.2. For an integer n ≥ 2, we have the following inequalities. Y 1 Y 1 (i) 1 ≤ 1 + < β(n) ≤ 1 + < A. p2 − 1 p(p − 1) p|n p|n (ii) 1 = lim inf β(n) < β(n) < lim sup β(n) = A. n−→∞ n−→∞ Average order in cyclic groups 109 Proof. We have Y 1 Y 1 − p−1 + p−2 1 + = p(p − 1) 1 − p−1 p prime p prime Y 1 − p−6 ζ(2)ζ(3) = = = A. (1 − p−2)(1 − p−3) ζ(6) p prime Claim (i) now follows from the multiplicativity of β and the lemma. For (ii), we clearly have 1 < β(n) < A for all n ≥ 2. When n ranges through 1 the primes, then β(n) = 1 + n(n−1) tends to 1, and when nk is the product of the first k primes, then limk→∞ β(nk) = A. 1.8 1.6 1.4 1.2 1 0 200 400 600 800 1000 Figure 2.1. Relative average order β(n) for n ≤ 1000. Figure 2.1 shows the behavior of β(n) for n ≤ 1000. The visible bands at 1 = β(1), 1.5 = β(2), 1.17 ≈ β(3), for example, are created by numbers of the form n = kp with small k and p either prime or having only large prime factors, namely k = 1, 2, 3 for the bands mentioned. We have seen that α(n) is firmly wedged between ϕ(n) and A · ϕ(n). Since lim infn→∞ ϕ(n)/n = 0, we also have lim inf α(n)/n = 0. n→∞ Theorem 4.4 below shows that this lower limit is even obtained on subse- quences corresponding to the multiplicative groups of finite fields. 110 J. von zur Gathen, A. Knopfmacher, F. Luca, L. Lucht, I. Shparlinski Our upper limit A occurs in several other contexts. Kendall & Rankin (1947), Section 3, consider the number of divisors of n that are divisible by the squarefree part of n, and show that its asymptotic mean value is A. Knopfmacher (1973) gives a more precise description of the mean value, and Knopfmacher (1972), Theorem 3.1 (vi), presents a generalization. The moments of this function are studied in Knopfmacher & Ridley (1974), Theorem 4.4. LeVeque (1977), Problem 6.5, determines A as the sum given in (3.7) below, and shows that the asymptotic mean of 1/ϕ is Ax−1 log x. The constant A also appears in Bateman (1972), Montgomery (1970), and Riesel & Vaughan (1983). Throughout the paper, log x is the natural logarithm of x. Section 3. The mean average order In this section, we determine the mean of the averaging functions α and β, and of γ = 1/β. A pleasant feature, due to double averaging, is that the error terms become rather small. We denote the average of an arithmetic function g byg ¯—not to be confused with complex conjugation: 1 X g¯(x) = g(n) x 1≤n≤x for x ≥ 1. There is a well-developed theory with many general results about the existence of means of arithmetic functions, see Elliott (1985); Indlekofer (1980, 1981); Postnikov (1988). However, those general results do not imply the specific statements of this work. The averageα ¯ is connected to the constant ζ(3) 3ζ(3) C = = ≈ 0.36538 14847. α 2ζ(2) π2 Theorem 3.1. The mean α¯ of α satisfies 2/3 4/3 α¯(x) = Cαx + O((log x) (loglog x) ) for x ≥ 3. Proof. We have 1 X 1 X 1 X 1 X 1 X α¯(x) = α(n) = dϕ(d) = ϕ(d). x x n x k 1≤n≤x 1≤n≤x d|n 1≤k≤x d≤x/k Walfisz (1963), Chapter IV, proves via exponential sum estimates that x ϕ¯(x) − ≤ c (log x)2/3(loglog x)4/3 for x ≥ 3, 2ζ(2) with some constant c. Now from X 1 Z ∞ dt 1 ≤ = k3 t3 2(x − 1)2 x<k x−1 Average order in cyclic groups 111 we obtain 2 1 X 1 X 1 X 1 x |α¯(x) − Cαx| = ϕ(d) − · x k x k 2ζ(2)k2 1≤k≤x 1≤d≤x/k 1≤k≤x 2 1 X 1 x ζ(3)x + · − x k 2ζ(2)k2 2ζ(2) 1≤k≤x 1 X 1 x x X 1 < · · c (log x)2/3(loglog x)4/3 + x k k 2ζ(2) k3 1≤k≤x x<k x ≤ c ζ(2)(log x)2/3(loglog x)4/3 + . 4ζ(2)(x − 1)2 Montgomery (1987) has shown that the error in the estimate forϕ ¯(x) is not O((loglog x)1/2), and conjectured that its maximum order is loglog x. We also have an explicit but worse error bound, both forϕ ¯ and forα ¯. Lemma 3.2. For x ≥ 1, we have x (i) ϕ¯(x) − < 2 + log x, 2ζ(2) (ii) α¯(x) − Cαx < 4 + ζ(2) log x. Proof. It is easily verified that (i) holds for 1 ≤ x < 2. We let x ≥ 2, and observe that X jxk µ(d) = 1 d 1≤d≤x for x ≥ 1, see (Apostol 1976, Theorem 3.12). It follows that 1 X jxk2 jxk 1 X jxk2 1 ϕ¯(x) = µ(d) + = µ(d) + 2x d d 2x d 2x 1≤d≤x 1≤d≤x x X µ(d) 1 X x jxk x jxk 1 = − µ(d) − + + .