ON A PROBLEM OF ARNOLD: THE AVERAGE MULTIPLICATIVE ORDER OF A GIVEN INTEGER

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1 ON A PROBLEM OF ARNOLD: THE AVERAGE MULTIPLICATIVE ORDER OF A GIVEN INTEGER Abstract. For g, n coprime integers, let l g n denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l g n as n ranges over integers coprime to g, and tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l g p as p ranges over primes.. Introduction Given coprime integers g, n with n > 0 and g >, let l g n denote the multiplicative order of g modulo n, i.e., the smallest integer k such that g k mod n. For an integer let T g := l g n, n n,g= essentially the average multiplicative order of g. In [], Arnold conjectured that if g >, then T g cg log, as, for some constant cg > 0. However, in [2] Shparlinski showed that if the Generalized Riemann Hypothesis GRH is true, then T g log ep Cglog log log 3/2, Date: June 2, 202. P.K. was partially supported by grants from the Göran Gustafsson Foundation, the Knut and Alice Wallenberg foundation, the Royal Swedish Academy of Sciences, and the Swedish Research Council. C.P. was supported by NSF grant numbers DMS , DMS What is needed is that the Riemann hypothesis holds for Dedekind zeta functions ζ Kn s for all n >, where K n is the Kummer etension Qe 2πi/n, g /n.

2 2 where Cg > 0. He also suggested that it should be possible to obtain, again assuming GRH, a lower bound of the form as. Let B = e γ p T g log ep log log log 2+o, = , p 2 p + the product being over primes, and where γ is the Euler Mascheroni constant. The principal aim of this paper is to prove the following result. Theorem. Assuming the GRH, T g = B log log log ep + o log log log as, uniformly in g with < g log. implicit in this result holds unconditionally. The upper bound Let λn denote the eponent of the group Z/nZ. Commonly known as Carmichael s function, we have l g n λn when g, n =, so we immediately obtain that T g λn, and it is via this inequality that we are able to unconditionally establish the upper bound implicit in Theorem. Indeed, in [2], Erdős, Pomerance, and Schmutz determined the average order of λn showing that, as, 2 λn = n n B log log log ep + o. log log log Theorem thus shows under assumption of the GRH that the mean values of λn and l g n are of a similar order of magnitude. We know, on assuming the GRH, that λn/l g n is very small for almost all n e.g., see [4, 7]; in the latter paper Li and Pomerance in fact showed that λn/l g n log n olog log log n as n on a set of asymptotic density, so perhaps Theorem is not very surprising. However, in [2] it was also shown that the normal order of λn is quite a bit smaller

3 ON A PROBLEM OF ARNOLD 3 than the average order: there eists a subset S of the positive integers, of asymptotic density, such that for n S and n, n λn =, log n log log log n+a+log log log n +o where A > 0 is an eplicit constant. Thus the main contribution to the average of λn comes from a density-zero subset of the integers, and to obtain our result on the average multiplicative order, we must show that l g n is large for many n for which λn is large. We remark that if one averages over g as well, then a result like our Theorem holds unconditionally. In particular, it follows from Luca and Shparlinski [9, Theorem 6] that 2 n <g<n g,n= l g n = log ep B log log + o log log log as. We also note that our methods give that Theorem still holds for g = a/b a rational number, with uniform error for a, b log, and n ranging over integers coprime to ab... Averaging over prime moduli. We shall always have the letters p, q denoting prime numbers. Given a rational number g 0, ± and a prime p not dividing the numerator or denominator of g, let l g p denote the multiplicative order of g modulo p. For simplicity, when p does divide the numerator or denominator of g, we let l g p =. Further, given k Z +, let D g k := [Qg /k, e 2πi/k : Q] denote the degree of the Kummer etension obtained by taking the splitting field of X k g. Let radk denote the largest squarefree divisor of k and let ωk be the number of primes dividing radk. Theorem 2. Given g Q, g 0, ±, define φkradk ωk c g :=. k 2 D g k k= The series for c g converges absolutely, and, assuming the GRH, l g p = π 2 c g + O. log 2 4/ log log log Further, with g = a/b where a, b Z, the error estimate holds uniformly for a, b.

4 4 At the heart of our claims of uniformity, both in Theorems and 2, is our Theorem 6 in Section 2. Though perhaps not obvious from the definition, c g > 0 for all g 0, ±. In order to determine c g, define c := p = , p 3 p the product being over primes; c g turns out to be a positive rational multiple of c. To sum the series that defines c g we will need some further notation. For p a prime and α Q, let v p α be the eponential p- valuation at α, that is, it is the integer for which p vpα α is invertible modulo p. Write g = ±g h 0 where h is a positive integer and g 0 > 0 is not an eact power of a rational number, and write g 0 = g g 2 2 where g is a squarefree integer and g 2 is a rational. Let e = v 2 h and define g = g if g mod 4, and g = 4g if g 2 or 3 mod 4. For g > 0, define n = lcm[2 e+, g]. For g < 0, define n = 2g if e = 0 and g 3 mod 4, or e = and g 2 mod 4; let n = lcm[2 e+2, g] otherwise. Consider the multiplicative function fk = ωk radkh, k/k 3. We note that for p prime and j, fp j = p 3j+minj,vph. Given an integer t, define F p, t and F p by t F p, t := fp j, F p := j=0 j= fp j j=0 In particular, we note that if p h, then 3 F p = p 3j = p p 3. Proposition 3. With notation as above, if g < 0 and e > 0, we have c g = c F p F 2, e + p + F p, v pn, 2F 2 F p p 3 p n p h otherwise c g = c p h F p p p 3 + p n F p, v pn. F p

5 ON A PROBLEM OF ARNOLD 5 For eample, if g = 2, then h =, e = 0, and n = 8. Thus F 2, 3 c 2 = c + = c 2 2/2 3 2/2 2 3 = c 59 F 2 2/8 60. We remark that the universal constant c = p p is already p 3 present in the work of Stephens on prime divisors of recurrence sequences. Motivated by a conjecture of Laton, Stephens showed in [3] that on GRH, the limit lim l g p π p eists and equals c times a rational correction factor depending on g. In fact, from the result it is easy deduce our Theorem 2 with a somewhat better error term. However, Stephens only treats integral g that are not powers, the error term is not uniform in g, and, as noted by Moree and Stevenhagen in [0], the correction factors must be adjusted in certain cases. Theorem 2 might also be compared with the work of Pappalardi []. In fact, his method suggests an alternate route to our Theorem 2, and would allow the upper bound π l g p 2 c g + o, as, to be established unconditionally. The advantage of our method is that it avoids computing the density of primes where g has a given inde. Finally, Theorem 2 should also be contrasted with the unconditional result of Luca [8] that p l π p 2 g p = c + O/log κ g= for any fied κ > 0. By partial summation one can then obtain π p p g= l g p c as, 2 a result that is more comparable to Theorem 2.

6 6 2. Some preliminary results For an integer m 2, we let P m denote the largest prime dividing m, and we let P =. Given a rational number g 0, ±, we recall the notation h, e, n described in Section., and for a positive integer k, we recall that D g k is the degree of the splitting field of X k g over Q. We record a result of Wagstaff on D g k, see [4], Proposition 4. and the second paragraph in the proof of Theorem 2.2. Proposition 4. With notation as above, 4 D g k = φk k k, h ɛ g k where φ is Euler s function and ɛ g k is defined as follows: If g > 0, then { 2 if n k, ɛ g k := if n k. If g < 0, then 2 if n k, ɛ g k := /2 if 2 k and 2 e+ k, otherwise. We also record a GRH-conditional version of the Chebotarev density theorem for Kummerian fields over Q, see Hooley [3, Sec. 5] and Lagarias and Odlyzko [6, Theorem ]. Let i g p = p /l g p, the inde of g in Z/pZ when g Z/pZ. Theorem 5. Assume the GRH. Suppose g = a/b 0, ± where a, b are integers of absolute value at most. For each integer k, we have that the number of primes p for which k i g p is D g k π + O/2 log. Note that k i g p if and only if k g splits completely modulo p. Also note that the trivial bound /k is majorized by the error term in Theorem 5 when k /2 / log. In fact, the error term majorizes the main term for k /4. We will need the following uniform version of [5, Theorem 23]. Theorem 6. If the GRH is true, then for, L with L log and g = a/b 0, ± where a, b are integers with a, b, we have { p : l g p p } π L L hτh log log + φh log 2

7 ON A PROBLEM OF ARNOLD 7 uniformly, where τh is the number of divisors of h. Proof. Since the proof is rather similar to the proof of the main theorem in [3], [4, Theorem 2], and [5, Theorem 23], we only give a brief outline. We see that l g p p /L implies that i g p L. Further, in the case that p ab, where we are defining l g p = and hence i g p = p, the number of primes p is Olog. So we assume that p ab. First step: Consider primes p such that i g p > /2 log 2. Such a prime p divides a k b k for some positive integer k < /2 / log 2. Since ω a k b k k log, it follows that the number of primes p in this case is O /2 / log 2 2 log = O/ log 3. Second step: Consider primes p such that q i g p for some prime q in the interval I := [ /2, log 2 /2 log 2 ]. We may bound this by considering primes p such that p mod q for some prime q I. The Brun Titchmarsh inequality then gives that the number of such primes p is at most a constant times q I φq log/q log q q I log log log 2. Third step: Now consider primes p such that q i g p for some prime q in the interval [L, /2. In this range we use Proposition 4 and log 2 Theorem 5 to get on the GRH that q [L, /2 log 2 {p : q i g p} πq, h qφq + /2 log. Summing over primes q, we find that the number of such p is bounded by a constant times πq, h + /2 log πωh q 2 L + log 2. Fourth step: For the remaining primes p, any prime divisor q i g p is smaller than L. Hence i g p must be divisible by some integer d in the interval [L, L 2 ]. By Proposition 4 and Theorem 5, assuming the GRH, we have πd, h 5 {p : d i g p} 2 + O /2 log. dφd Hence the total number of such p is bounded by πd, h 2 + O /2 log π hτh dφd L φh, d [L,L 2 ]

8 8 where the last estimate follows from d, h 6 dφd m dφd m h m h d [L,L 2 ] m h d [L,L 2 ] m d m Lφm = h Lφh m h k L/m φmkφk m φm φh h hτh Lφh. Here we used the bound k T /T for T > 0, which follows kφk by an elementary argument from the bound k T /T and the k 2 identity k/φk = µ 2 j j k. Indeed, φj k T kφk = j µ 2 j φjj 2 l T/j l 2 T j T φjj + j>t j 2 T. Corollary 7. Assume the GRH is true. Let m 2 be an integer and 0 7 a real number. Let y = log log and assume that m log y/ log log y. Let g = a/b 0, ± where a, b are integers with a, b eplog 3/m, and let h be as above. Then uniformly, p y m +. q P i gp>m q h, q>m Proof. This result is more a corollary of the proof of Theorem 6 than its statement. We consider intervals I j := e j, e j+ ] for j log, j a non-negative integer. The sum of reciprocals of all primes p eplog /m is y/m + O, so this contribution to the sum is under control. We thus may restrict to the consideration of primes p I j for j > log /m. For such an integer j, let t = e j+. If q i g p for some prime q > t /2 log 2 t, then l g p t /2 / log 2 t, and the number of such primes is O k t /2 / log 2 t k log ab = Ot log ab / log4 t, so that the sum of their reciprocals is Olog ab / log 4 t = Olog 3/m /j 4. Summing this for j > log /m, we get O, which is acceptable. For J := t /2 / log 2 t, t /2 log 2 t], with t = e j+, we have that the reciprocal sum of the primes p I j with some q J dividing i g p so that q p is Olog log t/ log 2 t = Olog j/j 2. Summing this for j > log /m is o as and is acceptable. For q t /2 / log 2 t we need the GRH. As in the proof of Theorem 6, the number of primes p I j with q i g p is bounded by a constant

9 ON A PROBLEM OF ARNOLD 9 times t q, h + t /2 log t. log t q 2 Thus, the reciprocal sum of these primes p is q, h O q 2 log t + log t q, h = O t /2 q 2 j + j. e j/2 We sum this epression over primes q with m < q e j/2 /j 2 getting O jm log m + j q +. j 2 q h, q>m Summing on j log completes the proof. 3. Proof of Theorem Let be large and let g be an integer with < g log. Define y = log log, m = y/ log 3 y, D = m!, and let S k = {p : p, D = 2k}. Then S, S 2,..., S D/2 are disjoint sets of primes whose union equals {2 < p }. Let { p } 7 Sk = p S k : p g, l g p 2k be the subset of S k where l g p is large. Note that if k log y, p S k \ S k, and p g, there is some prime q > m with q p /l g p, so that P i g p > m. Indeed, for sufficiently large, we have log y m/2, and thus k log y implies that each prime dividing D also divides D/2k, so that p, D = 2k implies that the least prime factor of p /2k eceeds m. Thus, from Theorem 6, S k \ S k {p : l g p < p/m} + p g π m hτh φh uniformly for k log y. Using this it is easy to see that S k and S k are of similar size when k is small. However, we shall essentially measure the size of S k or S k by the sum of the reciprocals of its members and for this we will use Corollary 7. We define E k := p S k <p α p α

10 0 and Ẽ k := p Sk <p α p α. By Lemma of [2], uniformly for k log 2 y, 8 E k = y log y P k + o where 9 P k = e γ k q>2 Note that, with B given by, P k 0 2k = B. k= q q 2 q 2. q k, q>2 The following lemma shows that not much is lost when restricting to primes p S k. Lemma 8. For k log y, we uniformly have log 5 y Ẽ k = E k + O. y Proof. By 8 and 9, we have E k y k log y y log 2 y, and it is thus sufficient to show that p S k \ S k /p log 3 y since the contribution from prime powers p α for α 2 is O. As we have seen, if k log y and p S k \ S k then either p g or P i g p > m. Hence, using Corollary 7 and noting that the hypothesis g log implies that h y and so h has at most one prime factor q > m, we have p E k \Ẽk This completes the proof. p y m = y y/ log 3 y log3 y. Lemma 9. We have k log y where B is given by. E k 2k = By + o log y Proof. This follows immediately from 8, 9, and 0.

11 ON A PROBLEM OF ARNOLD Given a vector j = j, j 2,..., j D/2 with each j i Z 0, let j := j + j j D/2. Paralleling the notation Ω i ; j from [2], let: Ω ; j be the set of integers that can be formed by taking products of v = j distinct primes p, p 2,..., p v in such a way that: for each i, p i < /y3, and the first j primes are in S, the net j 2 are in S 2, etc.; Ω 2 ; j be the set of integers u = p p 2 p v Ω ; j such that p i, p j divides D for all i j; Ω 3 ; j be the set of integers of the form n = up where u Ω 2 ; j and p satisfies p, D = 2, ma/2u, /y < p /u and l g p > p/y 2 ; Ω 4 ; j be the set of integers n = p p 2 p v p in Ω 3 ; j with the additional property that p, p i = 2 for all i. In the third bullet, note that the ma is not strictly necessary since when is sufficiently large, /2u > /y. 3.. Some lemmas. We shall also need the following analogues of Lemmas 2-4 of [2]. Let J := {j : 0 j k E k /k for k log y, and j k = 0 for k > log y}. Lemma 0. If j J, n Ω 4 ; j, and, then l g n c 2k j k, y 3 k log y where, c > 0 are absolute constants. Proof. Suppose that n = p p 2 p v p Ω 4 ; j. Let d i = p i, D, and let u i := p i /d i. By 7, u i divides l g p i for all i, and by the definition of Ω 3 ; j we also have l g p > p/y 2. Since p /2 is coprime to p i /2 for each i and each p i, p j D for i j, we have u,..., u v, p pairwise coprime. But l g n = lcm[l g p, l g p 2,..., l g p v, l g p], so we find that, using the minimal order of Euler s function and l g p > p/y 2, φn l g n u u 2 u v l g p y 2 v i= d i n y 2 log log n l k= 2kj k y 3 l k= 2kj k

12 2 recalling that d i = p i, D = 2k if p i S k, and that n Ω 4 ; j implies that n > /2. Lemma. If j J, u Ω 2 ; j, and 2, then {p : up Ω 4 ; j} > c 2 /uy log where 2, c 2 > 0 are absolute constants. Proof. Note that for j J, j l k= E k/k y/ log y by 8 and 9. For such vectors j, Lemma 3 of [2] implies that the number of primes p with ma/2u, /y < p /u, p, D = 2, and p, p i = 2 for all p i u is /uy log. Thus it suffices to show that {p /u : p, D = 2, l g p p/y 2 } = o/uy log. As we have seen, for j J, j y/ log y, so that u Ω 2 ; j has u /y2 for all large. Thus, Theorem 6 implies that π/u y 2 uy 2 log = o. uy log /u l gp p/y 2 The result follows. Lemma 2. If j J, then for 3, u > ep c3 y log log y log 2 y u Ω 2 ;j where 3, c 3 > 0 are absolute constants. Proof. The sum in the lemma is equal to j!j 2! j log y! p,p 2,...,p v k log y p p 2 p v where the sum is over sequences of distinct primes where the first j are in S, the net j 2 are in S 2, and so on, and also each p i, p j D for i j. Such a sum is estimated from below in Lemma 4 of [2] but without the etra conditions that differentiate S k from S k. The key prime reciprocal sum there is estimated on pages to be log log y E k + O. log y In our case we have the etra conditions that p g and p /2k l g p, which alters the sum by a factor of + Olog 5 y/y by Lemma 8. But the factor + Olog 5 y/y is negligible compared with the factor E j k k j k!

13 ON A PROBLEM OF ARNOLD 3 + Olog log y/ log y, so we have eactly the same epression in our current case. The proof is complete Conclusion. For brevity, let l = log y. We clearly have T g l g n. By Lemma 0, we thus have Now, T g y 3 n Ω 4 ;j and by Lemma, this is j J n Ω 4 ;j j J k= = l 2k j k n Ω 4 ;j u Ω 2 ;j up Ω 4 ;j u Ω 2 ;j uy log, which in turn by Lemma 2 is l y log ep c3 y log log y log 2 y Hence T g y 4 log ep c3 y log log y log 2 y Now, j J k= l 2k j Ej k k k = j k! l k= [E k /k] j k =0, k=. j J k= E j k k j k!. l 2k j Ej k k k j k!. E k /2k j k. j k! Note that 2w j=0 wj /j! > e w /2 for w and also that E k /2k for sufficiently large, as E k y/k log y by. Thus, j J k= l 2k j Ej k l k k > 2 l ep j k! Hence T g l y 4 log ep c3 y log log y log 2 2 l E k ep. y 2k k= E k 2k. k=

14 4 By Lemma 9 we thus have the lower bound in the theorem. The proof is concluded. 4. Averaging over prime moduli the proofs 4.. Proof of Theorem 2. Let z = log and abbreviate l g p, i g p with lp, ip, respectively. We have lp = ip z lp + ip>z lp = A + E, say. Writing lp = p /ip and using the identity /ip = µv/u, we find that uv ip A = p ip z uv ip = p = A E, say. The main term A is uv ip uv z A = uv z µv u µv u µv u p µv u uv ip ip>z uv z p. uv ip By a simple partial summation using Theorem 5, the inner sum here is Li 2 D g uv + O 3/2 log assuming the GRH. Thus, A = Li 2 µv + O 3/2 log ud uv z g uv n z uv=n µv u. The inner sum in the O-term is bounded by φn/n, so the O-term is O 3/2 log 2. Recalling that radn denotes the largest squarefree divisor of n, we note that v k µvv = p k p = ωk φradk, and hence µv ud g uv = µvv D g kk = ωk φradk. D g kk uv=k v k

15 ON A PROBLEM OF ARNOLD 5 On noting that φradk = φkradk/k, we have µv ud u,v g uv = ωk radkφk = c D g kk 2 g. k Thus, with ψh := hτh/φh, µv uvd uv z g uv = c g ωk radkφk = c D g kk 2 g + Oψh/z, k>z by Proposition 4 and the same argument as in the fourth step of the proof of Theorem 6 in particular, see 6. It now follows that A = Li 2 c g + Oψh/z. It remains to estimate the two error terms E, E. Using Theorem 6, we have E log log z log 2 ψh 2 ψh log 2. Toward estimating E, note that f z n := uv n uv z µv u d n v d d z Further, from the last sum we get f z n p a n p z µvv d = d n d z + p + + p p p a φradd d < 2 ωnz, z. where n z denotes the largest divisor of n composed of primes in [, z]. We have E p f z ip ip>z ip>z f z ip. Let w := 4 log z/ log log z. We break the above sum into three possibly overlapping parts: E, := f z ip, E,2 := f z ip, ip>z ωip z w E,3 := ip> /2 log 2 z<ip /2 log 2 ωip z>w f z ip.

16 6 Using Theorem 6, we have E, 2 w ip>z 2 w ψh 2 log log log 2. The estimate for E,3 is similarly brief, this time using the first step in the proof of Theorem 6. We have E,3 z ip> /2 log 2 The estimate for E,2 takes a little work. inequality, E,2 z 2 z<n /2 log 2 ωn z>w P m z ωm>w φm This last sum is smaller than k>w k! p z π; n, 2 z log n /2 log 2 2 log 2. By the Brun Titchmarsh z<n /2 log 2 ωn z>w φn 2 log k p + pp +... = k>w = k>w φn P m z ωm>w k! p z φm. k p p 2 k! log log z + Ok. The terms in this series are decaying at least geometrically by a large factor, so by a weak form of Stirling s formula, we have P m z ωm>w m ep w log log log z w log w + w + Ow/ log log z. By our choice for w, this last epression is smaller than ep 3 log z = log 3 for all large values of. Hence, E,2 2 log 2.

17 ON A PROBLEM OF ARNOLD 7 Noting that ψh τh log log, we conclude that lp = A + E = A + E + O E, + E,2 + E,3 = c g Li O log 2 ψh + 2w ψh log log + + = c g Li 2 + O 2 w τh 2 log log 2 = c g 2 π + O log 2 2, log 2 4/ log log log using that Li 2 = 2 π + O2 / log 2, the definition of w, and h log together with Wigert s theorem for the maimal order of the divisor function τh. This completes the proof Proof of Proposition 3. Proof of Proposition 3. We begin with the cases g > 0, or g < 0 and e = 0. Recalling that D g k = φkk/ɛ g kk, h, we find that 2 c g = k ωk radkφk D g kk 2 = k ωk radkk, hɛ g k k 3. Now, since ɛ g k equals if n k, and 2 otherwise, 2 equals 3 ωk radkh, k + ωk radkh, k = fk+fkn k 3 k 3 k n k k where the function fk = ωk radkh, k/k 3 is multiplicative. If p h and j, we have fp j = p/p 3j. On the other hand, writing h = p h pe h,p we have fp j = p +minj,e h,p /p 3j for p h and j. Since f is multiplicative, fk + fkn = fk + fkn k k : radk hn k,hn= fk. Now, for p h and j, we have fp j = radp j /p 3j = p/p 3j, hence j 0 fpj p = = p and thus p 3 /p 3 p 3 fk = F p = p p 3 = c k,hn= p hn p hn p hn p. p 3

18 8 Similarly, radk hn fk = p hn F p and fkn = fp j = F p F p, e n,p. radk hn p hn j e n,p p hn Hence Thus radk hn c g = fk + F p F p, e n,p fkn = F p + radk hn p hn p hn = F p + F p, e n,p. F p p hn p hn c p hn p F p + F p, e n,p, F p p 3 p hn p hn which, by 3, simplifies to c g = c p h F p p p 3 + p hn F p, e n,p. F p The case g < 0 and e > 0 is similar: using the multiplicativity of f together with the definition of ɛ g k, we find that c g = k = p = p fk + fkn 2 F p + p F p + p n e j= k,2= f2 j k F p F p, e n,p F 2, e + F p 2 p>2 F p, e n,p F 2, e +. F p 2F 2 Again using the fact that F p = p p h the proof is concluded. p p 3 + F p = c p h p h F p p/p 3 +

19 ON A PROBLEM OF ARNOLD 9 5. Acknowledgments Part of this work was done while the authors visited MSRI, as part of the semester program Arithmetic Statistics. We thank MSRI for their support, funded through the NSF. We are very grateful to Michel Balazard for suggesting Arnold s conjecture to us. In addition we thank Pieter Moree, Adam Feli, and the two anonymous referees for helpful comments. References [] V. Arnold. Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics. J. Math. Fluid Mech., 7suppl. :S4 S50, [2] P. Erdős, C. Pomerance, and E. Schmutz. Carmichael s lambda function. Acta Arith., 584: , 99. [3] C. Hooley. On Artin s conjecture. J. Reine Angew. Math., 225: , 967. [4] P. Kurlberg. On the order of unimodular matrices modulo integers. Acta Arith., 02:4 5, [5] P. Kurlberg and C. Pomerance. On the period of the linear congruential and power generators. Acta Arith, 92:49 69, [6] J. C. Lagarias and A. M. Odlyzko. Effective versions of the Chebotarev density theorem. In Algebraic number fields: L-functions and Galois properties, A. Fröhlich, ed. Academic Press, London and New York, 977, pp [7] S. Li and C. Pomerance. On generalizing Artin s conjecture on primitive roots to composite moduli. J. Reine Angew. Math., 556: , [8] F. Luca. Some mean values related to average multiplicative orders of elements in finite fields. Ramanujan J., 9-2:33 44, [9] F. Luca and I. E. Shparlinski. Average multiplicative orders of elements modulo n. Acta Arith., 094:387 4, [0] P. Moree and P. Stevenhagen. A two-variable Artin conjecture. J. Number Theory 852:29 304, [] F. Pappalardi. On Hooley s theorem with weights. Number theory, II Rome, 995. Rend. Sem. Mat. Univ. Politec. Torino 534: , 995. [2] I. E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete Contin. Dyn. Syst., 74:90 97, [3] P. J. Stephens. Prime divisors of second-order linear recurrences, I. J. Number Theory, 83:33 332, 976. [4] S. S. Wagstaff, Jr. Pseudoprimes and a generalization of Artin s conjecture. Acta Arith., 4:4 50, 982. Department of Mathematics, Royal Institute of Technology, SE Stockholm, Sweden address: kurlberg@math.kth.se Mathematics Department, Dartmouth College, Hanover, NH , U.S.A. address: carl.pomerance@dartmouth.edu