THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE

Transcription

1 THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE SHUGUANG LI AND CARL POMERANCE Abstract. For a atural uber, let λ() deote the order of the largest cyclic subgroup of (Z/Z). For a give iteger a, let N a (x) deote the uber of x coprie to a for which a has order λ() i (Z/Z). Let R() deote the uber of eleets of (Z/Z) with order λ(). It is atural to copare N a (x) with x R()/. I this paper we show that the average of N a(x) for a y is ideed asyptotic to this su, provided y exp((2 + ε)(log x log log x) /2 ), thus iprovig a theore of the first author who had this for y exp((log x) 3/4 ). The result is to be copared with a siilar theore of Stephes who cosidered the case of prie ubers.. Itroductio. Let be a atural uber. It was kow to Gauss that the ultiplicative group (Z/Z) is cyclic if ad oly if is ot divisible by two differet odd pries or divisible by four, except for = 4 itself. I particular, this holds wheever is prie. Whe (Z/Z) is cyclic, a geerator is called a priitive root. I geeral, let λ() be the expoet of (Z/Z), the axial order of ay eleet i the group. Followig Carichael [], we broade the defiitio of a priitive root to a eleet of (Z/Z) which has order λ(). There are various atural questios associated with these cocepts. () Let R() deote the uber of residues odulo which are priitive roots for. Thus, R()/ is the proportio of residues odulo which are priitive roots. What is R()/ o average, ad what is it o average for prie? (2) For a fixed iteger a, let N a (x) deote the uber of atural ubers x for which a is a priitive root, ad let P a (x) deote the uber of such which are prie. What is the asyptotic distributio of N a (x) ad P a (x)? (3) What is the average asyptotic behavior of N a (x) as a rus over a short iterval, ad what is it for P a (x)? We first review what is kow for the prie case. If p is prie, the ultiplicative group (Z/pZ) is cyclic of order p, ad so it follows that R(p) = ϕ(p ), where ϕ is Euler s fuctio. Oe has (see Stephes [3, Lea ]) R(p) A as x, () π(x) p x p where A = p ( ) = p(p ) [MATHEMATIKA, 55 (2009) 67 76]

2 68 S. LI AND C. POMERANCE is kow as Arti s costat. This suggests that typically we should have P a (x) Aπ(x). It is easy to see though that for soe choices of a this caot hold, aely, for a a square or a =, sice for each such a there are at ost two pries for which a is a priitive root. Arti s cojecture is the assertio that for all other values of a there are ifiitely ay pries for which a is a priitive root ad, i fact, there is a positive ratioal c a with P a (x) c a Aπ(x). This cojecture was proved by Hooley [2] uder the assuptio of the geeralized Riea hypothesis. For surveys, see Li ad Poerace [7], Moree [] ad Murty [2]. Cocerig the third questio, Stephes [3] has show ucoditioally that, if y > exp(4(log x log log x) /2 ), the y a y P a (x) Aπ(x) as x. (2) Turig to the coposite case, the first author i [5] showed that (/x) x R()/ does ot ted to a liit as x. We have x x R() x log log log x ad li sup x R() R() > 0, li if = 0. x x x x x Let E deote the set of itegers a which are a power higher tha the first power or a square ties a eber of {±, ±2}. It was show by the first author i [4] that for a E we have N a (x) = o(x), ad that for every iteger a we have li if x N a (x)/x = 0. I [8] we showed that, uder the assuptio of the geeralized Riea hypothesis, for each iteger a E we have li sup x N a (x)/x > 0. To coplete our brief review of the literature, the first author showed i [6] that, for y exp((log x) 3/4 ), y a y N a (x) x R() as x. (3) The goal of this paper is to iprove the rage for y i (3) to a rage for y siilar to that i (2). We use siilar ethods to those already used i these probles. Let L(x) = exp((log x log log x) /2 ). We prove the followig theore. THEOREM. For y L(x) 8, y a y N a (x) = x R() ( ) x + O y /7.

3 ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM 69 Further, for ay fixed ε > 0 ad L(x) 2+ε y L(x) 8, N a (x) = ( R() x L(x) /2+ε/6 + O y a y x y /4 + x log x ) y 5/32. I particular, (3) holds i the rage y L(x) 2+ε. We reark that our proof ca be adapted to the case of P a (x), ad so allows a iproveet of (2) to the rage y L(x) 2+ε. 2. Preliiaries. Variables p, q always deote pries. For a positive iteger, we write p a if p a ad p a+. I this case, we also write v p () = a. The uiversal expoet fuctio λ() ca be coputed fro the prie factorizatio of as follows: λ() = lc{λ(p a ) : p a }, where λ(p a ) = ϕ(p a ) uless p = 2, a 3, i which case λ(2 a ) = 2 ϕ(2a ) = 2 a 2. For each prie q λ() (which is equivalet to the coditio q ϕ()) let D q () = {p a : v q (λ(p a )) = v q (λ())}. If v q (λ()) = v > 0, let q () deote the uber of cyclic factors C q v i (Z/Z), so that q () = #D q (), except i the case q = 2 ad 2 3 D 2 (), whe 2 () = + #D 2 (). The (see [5, 0]), R() = ϕ() ( q q() ). (4) q ϕ() Let rad() deote the largest square-free divisor of. Let E() = {a od : a λ()/ rad(λ()) (od )}, so that E() is a subgroup of (Z/Z). We say that a character χ od is eleetary if it is trivial o E(). Clearly the order of a eleetary character is square-free. For each square-free uber h ϕ(), let ρ (h) be the uber of eleetary characters od of order h. It is ot hard to see that ρ (h) = q h (q q() ). (5) For a character χ od, let c(χ) = ϕ() b χ(b), where idicates that the su is over priitive roots od i [, ]. Further, let { /ρ (ord χ) if χ is eleetary, c(χ) = 0 if χ is ot eleetary.

4 70 S. LI AND C. POMERANCE PROPOSITION 2. If χ od is a character, the c(χ) c(χ). Proof. Various eleets of the proof are i [6]; we give a self-cotaied proof here. To see that c(χ) = 0 for χ ot eleetary, ote that the priitive roots od coprise a uio of soe of the cosets of the subgroup E() i (Z/Z), so that we ca factor a E() χ(a) out of the character su b χ(b). This factor is zero uless χ is trivial o E(); that is, c(χ) = 0 for χ ot eleetary. Suppose ow that χ is eleetary. For each prie q ϕ(), let S q () be the q-sylow subgroup of (Z/Z). This group has expoet q v q(λ()) ; let R q () deote the set of ebers with this order. The a residue b od is a priitive root od if ad oly if it is of the for q ϕ() b q, where each b q R q (), ad, if it has such a represetatio, the it is uique. Thus, ϕ()c(χ) = b od χ(b) = ( ) χ(b q ). q ϕ() b q R q () The ier character su is #R q () if q ord χ, sice i this case χ acts as the trivial character o S q (). Suppose that q ord χ. Sice S q ()\R q () E() ad χ is eleetary, χ(b) = χ(b) χ(b) = 0 χ(b) b R q () b S q () b S q () b R q () = (#S q () #R q ()). b S q () b R q () We have #R q () = #S q ()( q q() ), ad so we coclude that b R q () χ(b) = Thus, usig (4) ad (5), we have ϕ()c(χ) = = q ord χ { #S q ()q q() if q ord χ, #S q ()( q q() ) if q ord χ. #S q () q q() q ϕ() q ord χ #S q ()( q q() ) ( ) ω R() (q q() ) = ( )ω R() ρ (ord(χ)), q ord χ where ω is the uber of pries dividig ord(χ). The propositio ow follows sice R() ϕ(). PROPOSITION 3. Suppose that k, d are coprie positive itegers ad that ψ is a eleetary character od kd that is iduced by a character. Each of the followig holds:

5 ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM 7 (i) v q (λ(k)) = v q (λ(kd)) for each q ord ψ; (ii) χ is eleetary; (iii) c(χ) c(ψ). Proof. Let h = ord ψ = ord χ, let q h, let v = v q (λ(k)), ad let w = v q (λ(kd)). Clearly, v w. Sice χ has order h, there is soe iteger a with χ(a) ad a qv (od k) for soe v v. Sice k, d arecoprie, there is a iteger b with b a (od k) ad b (od d). The b qv (od kd) ad ψ(b) = χ(a). Sice ψ is eleetary, it follows that b E(kd), so that v > w. Thus, we have v v w, which copletes the proof of (i). Suppose that χ is ot eleetary, so that χ is ot trivial o E(k). This the iplies that there is soe a E(k) with χ(a). As above, there is soe b with b a (od k) ad b (od d). Sice λ(k)/ rad(λ(k)) divides λ(kd)/ rad(λ(kd)), it follows that b E(kd). However, ψ(b) = χ(a), cotradictig the assuptio that ψ is eleetary. This proves (ii). Usig (i) ad k, d coprie we iediately have q (k) q (kd) for each q h, so that (5) iplies that ρ k (ord χ) ρ kd (ord ψ). Thus, (iii) follows fro (ii) ad Propositio The proof. Our startig poit is a lea fro [6]. Let X() deote the set of o-pricipal eleetary characters od, ad let S (x,y) = c(χ) χ(a). x χ X() a y It is show i [6] that a y N a (x) = y x R() + S (x,y) + O(x log x). (6) Thus, we would like to show that S (x,y) is sall. A atural thought is to use character su estiates to ajorize the su of χ(a), but to do this, it will be coveiet to deal with priitive characters. Let χ 0, deote the pricipal character od ad let deote a su over o-pricipal priitive characters. We have S (x,y) = x = x k χχ 0, X() k c(χχ 0, ) a y χ(a)χ 0, (a) c(χχ 0, ) d (d,k)= χ(d)µ(d) χ(a), a y/d

6 72 S. LI AND C. POMERANCE where we ca drop the coditio χχ 0, X() sice, if χχ 0, is ot eleetary, the Propositio 2 iplies that c(χχ 0, ) = 0. Thus, S (x,y) µ(d) c(χχ 0,dk ) d x a y/d say. We have S d = k x/d (k,d)= k x/d (k,d)= k x/d (k,d)= = µ(d) S d, (7) d x x/dk rad( ) k x/dk rad( ) k k x/d 2 x/dk ( 2,k)= 2 x/dk ( 2,k)= x dk c(χχ 0,dk 2 ) a y/d a y/d c(χχ 0,k ) c(χχ 0,k ), (8) a y/d where the first iequality follows fro Propositios 2 ad 3. We ow give a estiate that will be useful i the cases with d large. To do this, we trivially ajorize the character su a y/d χ(a) with y/d, so that S d xy d 2 k x/d k The su over ad χ is estiated as follows. LEMMA 4. If k is a positive iteger, the where τ is the divisor fuctio. c(χχ 0,k ) c(χχ 0,k ). k ϕ(k) τ(ϕ(k)), Proof. For each h ϕ(k), cosider those priitive characters of order h. The uber of the for which χχ 0,k is a eleetary character with odulus k is at ost ρ k (h). Hece, the cotributio to the ier su for each h is at ost oe, so that the ier su is ajorized by τ(ϕ(k)). Further, the su o of /, which is a ifiite su, has a Euler product ad is see to be k/ϕ(k). Thus, the lea follows. Usig Lea 4, we have S d xy d 2 k x/d τ(ϕ(k)). (9) ϕ(k)

7 ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM 73 We deduce fro [9] that τ(ϕ()) x exp(c(log x/log log x) /2 ) (0) x for ay fixed c > 8/e γ = Usig this result, the estiate /ϕ(k) (log log x)/k for k x, ad partial suatio, we obtai fro (9) that S d xy d 2 exp(3(log x/log log x)/2 ). () We use this estiate whe d is large. For a positive iteger k ad positive reals w, z, let F(k, z) = c(χχ 0,k ), Note that T (w, z) = F(k, z), k w S(w, z) = w F(k, z). k k w a z S d S(x/d, y/d). (2) We ow look to estiate S(w, z) ad, to do this, we first estiate T (w, z) so that a partial suatio calculatio will give us S(w, z). LEMMA 5. For w, z 3 ad z L(w) 6, uiforly, Further, if L(w) 8 z L(w) 2, the as w, T (w, z) wz 3/6 L(w) /4. (3) T (w, z) wz 3/4 L(w) /2+o(). (4) Proof. We first cosider the case whe w z 3/2. We have, by the Pólya Viogradov iequality (see [3, Theore 2.5]), T (w, z) k /2 log k k w Usig Lea 4, we have T (w, z) k w c(χχ 0,k ). k 3/2 log k τ(ϕ(k)) w 3/2 log w ϕ(k) ϕ(k) τ(ϕ(k)). k w Thus, usig the sae arguet that allowed us to deduce () fro (9), we have T (w, z) w 3/2 exp(3(log w/log log w) /2 ).

8 74 S. LI AND C. POMERANCE Sice w 3/2 wz 3/4 whe w z 3/2, the lea follows i this case. Now assue that w > z 3/2. We use Hölder s iequality. Let r be a positive iteger, so that writig / as / (2r )/2r / /2r, where B = k w A = k w T (w, z) 2r A 2r B, (5) a z 2r c(χχ 0,k ) 2r/(2r ), = k ϕ(k) k w Usig 0 c(χχ 0,k ), Lea 4 ad (0), we have a z A w exp(3(log w/log log w) /2 ). (6) To estiate B we use the large sieve (see [3, Theore 7.3]) ad [3, Leas 3, 4 ad 5], ad obtai B (w 2 + z r )z r (r log z) r 2 uiforly for itegers r ad ubers w 3, z 3. We let so that w 2 z r, which iplies that r = 2 log w/log z, B z 2r (r log z) r 2. Further, r < 2 log w/log z + < (8/3) log w/log z, usig w > z 3/2. Thus, r log z < 3 log w. We coclude that ( ) r B /2r z(r log z) r/2 < z exp log(3 log w) 2 ( ) 4 log w log(3 log w) < z exp. (7) 3 log z Sice r < (8/3) log w/log z, we have w /2r > z 3/6, so that, fro (6), we have A (2r )/2r w z 3/6 exp(3(log w/log log w)/2 ). Usig this estiate with (5) ad (7), we have ( ( ) log w /2 ) T (w, z) wz 3/6 log w log(3 log w) exp 3 +, log log w (3/4) log z ad so the lea follows i the rage z L(w) 6. If z L(w) 8, the the above choice of r is (2 + o()) log w/log z, so that (6) iplies that A (2r )/2r wz /4 L(w) o(). Further, if z L(w) 2, the the first part of (7) shows that B /2r zl(w) /2+o(), so that the iequality T (w, z) wz 3/4 L(w) /2+o() follows fro (5). 2r.

9 ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM 75 We are ow ready to coplete the proof of the theore. Usig Lea 5 ad partial suatio, we deduce that if z L(w) 6, the S(w, z) w log w z 3/6 L(w) /4. (8) If L(w) 8 z L(w) 2, the usig S(w, z) = T (w, z) + w w u 2 T (u, z) du, we distiguish the cases L(u) 8 z ad L(u) 8 > z. I the first case, we use (3) ad L(u) /4 z /32 to obtai a upper boud of order log w z 27/32 for the itegral. I the secod case, we use (4) to obtai the upper boud z 3/4 L(w) /2+o() for the itegral. So, for ay fixed δ > 0, whe L(w) 8 z L(w) 2 we have S(w, z) wz 3/4 L(w) /2+δ + w log w z 27/32. (9) Suppose first that L(x) 8 y x. For d y /4, we have y/d y 3/4 L(x) 6 L(x/d) 6. Thus, fro (2) ad (8), S d S(x/d, y/d) xy log x d 29/6 y 3/6 L(x)/4. Suig this for d y /4 ad () for d > y /4, ad usig (6) ad (7), we have the theore i the case that y L(x) 8. Now let ε > 0 ad assue that L(x) 2+ε y L(x) 8. If d y/l(x) 2+ε/2, for large x we see that L(x/d) 8 L(x) 8 /d y/d L(x) 2+ε/2 > L(x/d) 2. Thus, by (2) ad (9) with δ = ε/6, we have S d S(x/d, y/d) xy xy log x d 7/4 y /4 L(x)/2+ε/6 + d 59/32 y 5/32. We su this for d y/l(x) 2+ε/2, add i the su of the estiate i () for larger d, ad obtai fro (7) that y S (x,y) x y /4 L(x)/2+ε/6 + x log x y 5/32 + x y L(x)2+ε/2+ε/6. Note that the first ter doiates the third i the give rage for y. We ow use (6) to coplete the proof of the theore. Ackowledgeet. The secod author was supported by NSF grat uber DMS Refereces. R. D. Carichael, The Theory of Nubers, Wiley (New York, 94). 2. C. Hooley, O Arti s cojecture. J. Reie Agew. Math. 225 (967), H. Iwaiec ad E. Kowalski, Aalytic Nuber Theory, Aerica Matheatical Society (Providece, RI, 2004). 4. S. Li, O extedig Arti s cojecture to coposite oduli. Matheatika 46 (999), S. Li, Arti s cojecture o average for coposite oduli. J. Nuber Theory 84 (2000), S. Li, A iproveet of Arti s cojecture o average for coposite oduli. Matheatika 5 (2004),

10 76 S. LI AND C. POMERANCE 7. S. Li ad C. Poerace, Priitive roots: a survey. I Nuber Theoretic Methods Future Treds (Developets Matheatics 8) (eds. C. Jia ad S. Kaeitsu), Kluwer Acadeic (Dordrecht, 2002), S. Li ad C. Poerace, O geeralizig Arti s cojecture o priitive roots to coposite oduli. J. Reie Agew. Math. 556 (2003), F. Luca ad C. Poerace, O the average uber of divisors of the Euler fuctio. Publ. Math. Debrece 70 (2007), G. Marti, The least prie priitive root ad the shifted sieve. Acta Arith. 80 (997), P. Moree, Arti s priitive root cojecture a survey. Preprit, 2004, ath.nt/ M. R. Murty, Arti s cojecture for priitive roots. Math. Itelligecer 0 (988), P. J. Stephes, A average result for Arti s cojecture. Matheatika 6 (969), Shuguag Li, Departet of Matheatics, Uiversity of Hawaii at Hilo, Hilo, HI , U.S.A., E-ail: shuguag@hawaii.edu Carl Poerace, Matheatics Departet, Dartouth College, Haover, NH , U.S.A. E-ail: carl.poerace@dartouth.edu MSC (2000): Priary, N37; Secodary, A07. Received o the st of Deceber, Accepted o the 9th of Jauary, 2009.