AN IMPROVEMENT OF ARTIN S CONJECTURE ON AVERAGE FOR COMPOSITE MODULI. 1. Introduction

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1 AN IMPROVEMENT OF ARTIN S CONJECTURE ON AVERAGE FOR COMPOSITE MODULI SHUGUANG LI. Itroductio Let q be a atural umber. Whe the multiplicative group Z/qZ) is a cyclic group, its geerators are called primitive roots. Note that the geerators are also elemets with the maximum order if Z/qZ) is cyclic. Thus, whe Z/qZ) is ot a cyclic group, we the call its elemet with the maximal possible order a primitive root, which was iitially itroduced by R. Carmichael []. Let a be a iteger ad x be ay positive real umber. I 927, Arti cojectured that the umber P a x) of the primes up to x, for which a is a primitive root, is proportio to the umber of primes up to x, amely πx). Although this cojecture has ot bee proved ucoditioally, may results [3, 4, 6] have bee achieved, favorig the cojecture from various perspectives. For a survey of Arti s cojecture, the reader may refer to Murty [2]. Amog the ucoditioal results of Arti s cojecture is Stephes theorem [4] o the average of P a x) that is relevat to the work of this paper: if y > exp4l x l l x) /2 ) the ) y ) x P a x) = A li x + O l D x where A = prime p ) ad D is a arbitrary costat larger tha. I obtaiig pp ) this estimate, Stephes itroduced a character sum to P a x). Note that all moduli i P a x) are primes ad every o-pricipal character modulo a prime is a primitive character. It does ot take too much effort for Stephes to apply results from large sieve to the character sum to achieve the above theorem. I the same paper, he also obtaied a estimate of y P ax) A li x) 2, which allows him to fid a upper boud for the umber of exceptios to the result i ). What would happe if the moduli are ot restricted to primes? Let N a x) be the umber of moduli up to x for which a is a primitive root. I view of Arti s cojecture, oe might cojecture that N a x) is proportio to [x], the umber of atural umbers up

2 2 SHUGUANG LI to x. O the cotrary, this cojecture was proved wrog. I [8] the author proved that, for ay y x, 2) lim sup y x N a x) = 0 ad lim if y x N a x) > 0. x xy x xy Based o the results i 2), oe may aturally guess that, for most itegers a, the idividual N a x)/x should behave the same way as the average i 2). Namely, lim N a x)/x = 0, ad lim N a x)/x > 0 x x for most of itegers a. The first cojecture is proved i [9], free of ay hypothesis for all itegers a. The secod cojecture is proved i [0] o assumptio of GRH. The set of exceptioal itegers a to the secod cojecture ca also be foud i [0]. I compariso with ), the iterval of a for averagig N a x) i 2) is too large. The purpose of the paper is to show that 2) still holds for smaller values of y. Ideed, by itroducig Dirichlet characters to N ax) ad usig the well-kow iequality of Pólya-Viogradov see [3] ad [6]), we ca prove THEOREM. For ay positive real umbers x, y 3, we have 3) N a x) = ) )) R) y + O x 3/2 /2 l x y exp, l l x x where R) is the umber of primitive roots modulo withi iterval [, ]. Sice the upper boud for the character sums i Pólya-Viogradov iequality is attaied very rarely [5], its applicatio i Theorem leaves a lot of room for more accurate estimatio. With work i this directio, we are able to prove THEOREM 2. If x e 3 ad y expl x) 3 4 ), the R) 4) y + O x exp N a x) = x 5 6 l x) 2 )), where R) is the same as i Theorem. Moreover, l x) /2 i the error term ca be replaced by l y) 2 / l x if y is i the iterval [ expl x) 3 4 ), x.0 2 ]. It has bee proved [8]) that o a ubouded set of x we have x some positive costat c, ad o aother ubouded set of x we have x Combiig these results with Theorem 2, we ca deduce COROLLARY 3. The statemet i 2) holds for y > expl x) 3 4 ). R) R) c x for ox).

3 IMPROVEMENT OF AN AVERAGE ORDER 3 The mai idea that leads to the proof of the two theorems is ispired from Stephes method i provig ). The author believes that the best lower boud for y, for which similar statemet as i Theorem 2 holds, would be somethig like the oe i Stephes theorem i ). The author is grateful to Carl Pomerace for ecouragig him to work o the problem ad makig valuable suggestios. 2. Ivolvemet of Characters i Estimatio of N a x) I the followig, let C be the set of complex umbers ad C be the set of ozero complex umbers. A homomorphism from Abelia group G to C is called a character of G. Let q be ay positive iteger ad χ be a Dirichlet character, which is a fuctio χ from Z to C such that i) χ+q) = χ) for ay iteger ; ii) χ) 0 if gcd, q) = ad χ) = 0 if otherwise; iii) χ m) = χ) χm) for ay itegers ad m. Due to its periodicity, χ iduces a fuctio χ from Z/qZ to C such that 5) χ ψ = χ where ψ is the atural homomorphism from rig Z to Z/qZ. χ is also a group character of Z/qZ). Coversely each such group character χ ca be exteded over Z/qZ by defiig χa) = 0 for those residue classes a where gcda, q). The, by equatio 5), we obtai a Dirichlet character χ modulo q. I this sese we will use the same otatio for a Dirichlet character ad its correspodig group character of Z/qZ) although they are actually related by equatio 5). LEMMA 4. Let χ be a character modulo q, ad cχ) = φq) b χb) where the prime meas that the sum is take over all primitive roots betwee ad q modulo q. The we have χ mod q {, if a is a primitive root mod q cχ) χa) = 0, otherwise. Proof: By the defiitio of cχ) we have 6) cχ) χa) = χa) χb) = φq) χ mod q χ mod q b b φq) χ mod q χab). If a is ot a primitive root the ab mod q for ay primitive root b. If a is a primitive root the so is a. Sice χ mod q χc) = 0 if c mod q see p.30 of [2]),

4 4 SHUGUANG LI the oly otrivial cotributio to the sum o the right side of 6) comes from the terms with b = a, which is χa a ) =. φq) We have the proved the lemma. χ mod q LEMMA 5. Let N a x) be the umber of moduli up to x for which a is a primitive root. The, for ay real umbers x, y, we have 7) N a x) = y x R) + x χ mod cχ)χa) + Ox l x), where R) is the umber of primitive roots modulo withi the iterval [, ], ad χ 0 is the pricipal character modulo. Proof: Let t a ) be a coutig fuctio of primitive roots which takes value if a is a primitive root for, ad 0 if otherwise. By the defiitio of N a x) ad Lemma 4, N ax) ca be writte as 8) t a ) = t a ) = cχ)χa) x x x χ mod = cχ 0 )χ 0 a) + cχ)χa). x x χ mod By the iclusio ad exclusio priciple, the umber of positive itegers up to y which are relatively prime to is give by [y] ] p + ] p,q or [y] φ) + O2ω) ), where ω) is the umber of distict prime factors of iteger. Note. We have that cχ 0 ) = R) φ) cχ 0 )χ 0 a) = x x = x R) φ) gcda,)= [ [y] p [ [y] p q ) R) φ) φ) y + O2ω) ) = y R) + Ox l x), x where the average order of 2 ω) i the last equatio is a well-kow result [5]). We have proved the lemma.

5 IMPROVEMENT OF AN AVERAGE ORDER 5 3. Estimate of cχ) ad Proof of Theorem I view of Lemma 5 a sharp boud of cχ) for ay o-pricipal character χ modulo q is critical for a good estimate of the error term. I [4], it is foud that cχ) /ordχ) where ordχ) deotes the order of χ. But whe the moduli are ot primes, this boud is ot good eough. Throughout this sectio, C m represets a cyclic group of order m ad χ is a character of a fiite Abelia group. Ay elemet of a fiite Abelia group with the maximal order is called a primitive root of the group. LEMMA 6. Let G be a fiite Abelia group ad χ be a o-pricipal character of G. The χb) = 0. Proof: See p. 254 of [7]. b G LEMMA 7. Suppose that q is a prime ad v is a atural umber. Let χ be a character of cyclic group C q v. Let C q v be the cyclic subgroup of C q v. The { 0, if ordχ) > q, χb) = q v, if ordχ) q. b C q v Proof: Let α be a geerator of C q v, ad η = χα). The η qv = χα qv ) =. Thus the order of η, which is equal to order ordχ), divides q v. O the other had, sice α q is a geerator of the subgroup C q v, we have b C q v χb) = 0 k<q v χα qk ) = If ordχ) > q, the η q ad the above sum is equal to 0 k<q v η qk. η qv η q = 0. If ordχ) q, the η qk = ad the above sum is equal to q v. We have proved the lemma. Let q be a prime ad G be a fiite Abelia q-group, which meas that the order of G is a power of q. By a well-kow fact about Abelia groups, G ca be writte as 9) G = G G H,

6 6 SHUGUANG LI where each of G i is a cyclic subgroup of G of order q v, ad H has o cyclic subgroup of order q v. The it is well-kow that ay character χ of G ca be factored as, 0) χ = χ χ χ H where χ i ad χ H are the correspodig characters of G i ad H, respectively. LEMMA 8. Let q be a prime ad G be a fiite Abelia q-group for which 9) holds. Let χ be a character of G for which 0) holds. If χ is ot the pricipal character, the { G /q χb) =, if each ordχ i ) q ad χ H = χ 0, 0, otherwise. primitive roots: b G If χ is the pricipal character, the primitive roots: b G χb) = G /q ). Proof: If χ is ot the pricipal character, by Lemma 6, we have χb) + χb) = χb) = 0. b G primitive roots: b G o primitive roots: b G Let s deote the secod sum, ad let G i be the cyclic subgroup of G i of order G i /q. The s = b i G i, b H H χ b ) χ b )χ H b H ) = ) χ i b) χ H b). i= b G b H i By Lemma 7, b G χ i b) = 0 if ordχ i > q, ad is equal to G i /q if ordχ i ) q. By i Lemma 6, b H χ Hb) = 0 if χ H χ 0, ad is obviously H if otherwise. Therefore, combiig these facts with the above equatio yields the first formula i the lemma. If χ is the pricipal character, the above aalysis still holds, which yields s = G /q. O the other had, b G χb) = G. Thus, by the first equatio of the proof, χb) = G s = G /q ). primitive roots: b G We have proved the lemma.

7 IMPROVEMENT OF AN AVERAGE ORDER 7 Let G ad χ be the same as i Lemma 8. From the lemma, we have see that the character sum has a o-trivial value if ad oly if ordχ i ) q for each i ad χ H is the pricipal character of H. We call such a character χ of G a special character. Now let us tur our attetio to Dirichlet characters. Let be a ozero iteger. For each prime divisor q of φ), let G q be the Sylow q-subgroup of Z/Z). By a well-kow result of group theory, ) Z/Z) q φ) G q. Each character χ of Z/Z) ca be factored as q φ) χ q, where χ q is the correspodig character of G q. If each such factor χ q is a special character, the χ is called a special character of Z/Z). I other words, a special character is a character whose order is square-free ad the factor χ H defied i equatio 0) for each χ q is the pricipal character of H. THEOREM 9. Suppose that is a ozero iteger ad χ is a Dirichlet character modulo. Let ρ d) deote the umber of special characters modulo of order d. If χ is ot a special character, the cχ) = 0. If χ is a special character, the cχ) ρ ordχ)). Proof: By the defiitio give i Lemma 4, φ) cχ) = primitive roots b Z/Z) χb). With respect to the factorizatio of Z/Z) i ), we ca write χ = q φ) χ q, where χ q is a character of the Sylow q-subgroup G q. The χb) = q φ) χ qb q ), where b q φ) b q is the isomorphism i ). Note that b is a primitive root mod if ad oly if each b q is a primitive root i G q. Thus, φ) cχ) = primitive roots b q G q,q φ) q φ) χ q b q ) = q φ) χ q b q ). b q G q primitive roots If χ is ot a special character, the oe of its factors χ q is ot a special character of G q. By Lemma 8, the correspodig factor i the factorizatio of φ) cχ) above is zero. Thus, cχ) = 0.

8 8 SHUGUANG LI If χ is a special character, the so is each χ q. Note that either χ q = χ 0, the pricipal character of G q, or ordχ q ) = q. By Lemma 8, { Gq /q χ q b q ) = q), if ordχ q ) = q, G q /q q) ), if χ q = χ 0, primitive roots: b q G q where q ) is the subscript i the factorizatio of G = G q i 9) see [9] for aother explaatio of the fuctio q )). Therefore, q φ) 2) φ) cχ) G q q = q) q φ) ordχ q )=q φ) q. q) q φ) ordχ q )=q Note that the deomiator above is greater tha the umber of special characters of Z/Z) of order ordχ), which is q φ) q q) ). Therefore the iequality ordχ q )=q claimed i the theorem is true. We have proved the theorem. Obviously, the deomiator of 2) is greater tha or equal to q φ) q = ordχ), ordχ q)=q if χ is a special character. Therefore, by 2) if χ is a special character ad by Theorem 9 if χ is ot a special character, we have the followig corollary, which is used i the derivatio of the major results i [4]. But it is ot good eough for our derivatio of Theorem 2. COROLLARY 0. Let χ be a character modulo. The cχ) ordχ). Let be a divisor of. Let χ be a primitive character modulo which iduces χ modulo. What we eed for the deductio of Theorem 2 is the relatio betwee cχ) ad cχ ). Complete uderstadig of this relatio has yet to be clear. However, we fid a alterative aswer to the questio i the ext lemma. If χ is a special character, the the umber of the primitive characters χ modulo, which iduce special characters modulo of the fixed order ordχ), is less tha or equal to ρ ordχ)), the umber of special characters modulo with order ordχ). Note that here ordχ) is a squarefree iteger ad cχ). By Theorem 9 agai, we have deduced the ext lemma. COROLLARY. Let be a divisor of. For ay primitive character χ modulo, let χ be the iduced character modulo. The, for ay positive umber t, we

9 have χ mod cχ) t IMPROVEMENT OF AN AVERAGE ORDER 9 χ mod cχ) d φ ) µd) = 2 ωφ )) where the prime meas that the sums are take over primitive characters, ad ωφ )) is the umber of distict prime factors of φ ). We will ed the sectio with a proof of Theorem. LEMMA 2 see [2, 3, 6]). There exists a absolute costat c such that χa) c l for ay real umber y ad ay o-pricipal character χ to the modulus. Moreover, c if χ is a primitive character. LEMMA 3 see []). Let τ) be the umber of divisors of iteger. estimate ) /2 )) ) l x l x τφ)) = x exp cx) + O l l x l l x x The the holds for large real umbers x where cx) is a umber i the iterval [ 8 e γ, 8e γ ], ad γ is the Euler costat. Proof of Theorem : Let E be the triple sum i Lemma 5. Obviously we have E = cχ) χa) cχ) χa) x. x χ mod χ mod By Lemma 2, for some absolute costat c, we have E c cχ) /2 l = c /2 l x x χ mod χ mod cχ). Note that, from Theorem 9, cχ) = 0 if χ is ot a special character modulo, ad cχ) /ρ ordχ) if χ is a special character modulo. Also ote that ordχ is a square-free divisor of φ) if χ is special. Thus cχ) µd) = 2 ωφ)), χ mod d φ)

10 0 SHUGUANG LI where ω) is the umber of distict prime divisors of, ad E c x By Lemma 3, 2 ωφ)) x exp x /2 l 2 ωφ)) c x /2 l x x 2 ωφ)). 8e γ ) ) /2 l x l l x where the ivolved costats are absolute. Therefore, ) ) /2 l x E x 3/2 exp, l l x x ) ) /2 l x l x exp, l l x where the ivolved costat is absolute. We have the proved the theorem. 4. Proof of Theorem 2 LEMMA 4 see [4]). Let r be ay atural umber ad x, y be positive real umbers. The χa) x 2 + y r )y { r ley r ) } r 2 x χ mod where meas that the summatio is over primitive characters oly, ad the ivolved costat is idepedet of r. LEMMA 5. Let r 2 be{ ay atural umber. Let x, y be real umbers such that x 3 ad y. If δ = mi r l y } l x, l y is i the iterval 0, ), we have r + 2 l x χa) y ley r ) ) max{,r2 } 3) x δ x χ mod where meas the same as i Lemma 4, ad the ivolved costat is idepedet of r. Proof: Note that if 2 the oly character is the pricipal character. Thus 4) = x x χ mod x χ mod χa) χa) + x 3 t 2 t χ mod χa) dt.

11 IMPROVEMENT OF AN AVERAGE ORDER By Lemma 4, x x χ mod χa) ) x + yr y r ley r ) ) r 2. x Let δ be a real umber i 0, ). If δ r + 2 l y l x, the δ l x l yr provided r 2. Also ote that the characters below are primitive. Thus, the costat c i Lemma 2 ca be take as, ad we have x δ x δ χa) dt r l ) dt t 2 t 2 By Lemma 4 agai, x 3 t χ mod 3 3 t χ mod x δ x δr l x δ ) dt x δr+) δ l x) x δr+) l y r ). t 2 x δ t 2 t χ mod t χa) 2 r dt x x δ t 2 t2 + y r )y r ley r )) r2 dt ) y r ley r ) ) r 2. x + yr { r l y Now let δ = mi l x, 2 r r + 2 l y }. The we have x y r /x δ ad x δr+) l x y /x δ. Therefore, if 0 < δ <, by combiig the above estimates i 4), we have χa) y ley r ) ) max{,r2 }. x δ x χ mod We have proved the lemma. COROLLARY 6. Let x, y, r ad δ be the same as i Lemma 5. The, for ay atural umber d such that x/d, y/d, we have χa) y ley r ) ) max{,r2 } 5) x δ x/d χ mod /d where meas the same as i Lemma 4, ad the ivolved costat is idepedet of r. x δ

12 2 SHUGUANG LI Proof: Suppose that d x ad d y. Obviously χa) x x/d which is, by Lemma 5, χ mod /d χ mod χa) /d y/d) x δ ley/d) r ) ) max{,r2 } which i tur is less tha or equal to the fuctio o the right side of 5). Ideed the fuctio o the right side of 5) is max{y r x, y r+)/r+2) }ley r )) max{,r2 } which is icreasig i y, so that its value at y is at least as big as its value at y/d. Therefore, we have proved the corollary., Proof of Theorem 2: By Theorem oe ca see that Theorem 2 is true for y > x 2 expl 2 x). Thus we oly eed to cosider the case where y x.0/2. I view of Lemma 5 to prove Theorem 2 we oly eed to obtai a good estimate of S = x cχ) χ mod χa). Let χ be the primitive character modulo which iduces χ modulo. Defiitely. The χa) = χ ) µd) = χ d)µd) χ a). y d / /d d,/ ) Also ote that each χ is uiquely determied by χ, ad. Thus S cχ) µd) χ a) x χ mod d / /d = cχ) µd) χ a) x χ mod d / /d = µd) cχ) χ a) d x χ mod x/d x d /d

13 IMPROVEMENT OF AN AVERAGE ORDER 3 where prime meas that the sum is over the primitive characters modulo. Usig Holder s iequality ad Corollary, we obtai S d x x d d x By Lemma 3, d x x/d x d x )) d 2ωφ x/d χ mod cχ) x )) d 2ωφ x d d x l x exp d x x/d l x l l x 8e γ d x x d x d ) 2 χ mod x χ mod a y d d χ a) /d xl x) 2 exp where γ is Euler s costat. Assume that the coditios of Corollary 6 are satisfied. The x d χ a) d x x/d χ mod /d χ a). l x l l x 8e γ ) 2 x y x δ ley r ) ) max{,r2 } x y ley r ) ) max{,r2 } l x, x δ which, combied with the previous estimate, yields or 6) S x δ y l x) 2 exp S xy x Now let us take δ 8e γ l x) 2 exp 8e γ ) ) l x 2 ley r ) ) max{,r 2 } l l x ) ) l x 2 ley r ) ) max{,r 2 }. l l x [ ].6 l x r = +. l y The r 4 sice y x.0/2. Next we claim that which is equivalet to r l y l x r + 2 l y l x, d x d,

14 4 SHUGUANG LI 7) r 2 r + 2 l x l y. Ideed this iequality follows from the defiitio of r ad the fact that l x/ l y 2/.0. It ca be verified first for l x/ l y i the iterval [2/.0, 2.5) where r = 4. Whe l x/ l y 2.5, sice r 2 /r + 2) is a icreasig fuctio ad r >.6 l x/ l y, the left-had side of 7) is > ) 2.6 l x l y.6 l x l y + 2 l x l y. The the umber δ i Corollary 6 is. Obviously 0 < δ <. Thus, we have l x cleared all the assumed coditios i the derivatio of 6). As x δ/ = y /r+2), the logarithm of the right-had side of 6) 8) l y ) l x + 2 l l x + r r 2 + l ley r ). 8e γ l l x r+2 l y The first term of 8) is less tha or equal to l y) 2 /3.03 l x). Ideed, this iequality follows from r l x/ l y, which ca be verified for l x/ l y i the iterval [2/.0, 2.5) where r = 4, ad for l x/ l y 2.5 we have that r +2 <.6 l x/ l y l x/ l y. For y expl x) 3/4 ) the expressio l y) 2 / l x is at least l x) /2. The secod ad third terms i 8) are clearly ol x) /2 ), ad so is the fourth for the stated rage for y. By 6) ad 8), we have S x y exp 5 6 ) l y) 2 l x for all x e 3 ad y with expl x) 3 4 ) y x.0 2. Thus, we have Theorem 2. Refereces [] R.D. Carmichael, The Theory of Numbers, Wiley, New York, 94. [2] H. Daveport, Multiplicative Number Theory, Spriger-Verlag, New York, [3] R. Gupta ad M. Ram Murty, A remark o Arti s cojecture, Ivet. Math ), [4] D.R. Heath-Brow, Arti s cojecture for primitive roots, Quart. J. Math. Oxford 2), 37986), [5] A. Hildebrad, Large Values of Character Sums, J. Number Theory, 29988), [6] C. Hooley, O Arti s cojecture, J. reie agew. Math ), [7] K. Irelad ad M. Rose, A classical itroductio to moder umber theory, 2d ed., Spriger- Verlag, New York, 990. [8] S. Li, Arti s cojecture o average for composite moduli, J. Number Theory ), [9] S. Li, O extedig Arti s cojecture to composite moduli, Mathematika, 46999),

15 IMPROVEMENT OF AN AVERAGE ORDER 5 [0] S. Li ad C. Pomerace, O geeralizig Arti s cojecture o primitive roots to composite moduli, to appear i J. reie agew. Math. [] F. Luca ad C. Pomerace, O the average umber of divisors of the Euler fuctio, preprit. [2] M.R. Murty, Arti s cojecture for primitive roots, Math. Itelligecer 0 988), o. 4, [3] G. Pólya, Über die Verteilug der quadratische Reste ud Nichtreste, Nachrichte Köigl. Ges. Wiss. Göttige 98), pp [4] P.J. Stephes, A average result for Arti s cojecture, Mathematika 6969), [5] G. Teebaum, Itroductio to aalytic ad probabilistic umber theory, Cambridge Uiversity Press, Cambridge, 995. [6] I.M. Viogradov, Über die Verteilug der quadratische Reste ud Nichtreste, J. Soc. Phys. Math. Uiv. Permi 299), pp. -4.