EXPLICIT CONSTANTS IN AVERAGES INVOLVING THE MULTIPLICATIVE ORDER

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1 EXPLICIT CONSTANTS IN AVERAGES INVOLVING THE MULTIPLICATIVE ORDER KIM, SUNGJIN DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES MATH SCIENCE BUILDING 667A Abstract Let a > Denote by l ap the multiplicative order of a modulo p We look for an estimate of sum of lap over primes p on average When e average over a N, e observe a statistic of CLi P J Stephens [S, Theorem ] proved this statistic for N > epc log for some positive constant c Upon this result, the value of c is at least e 9 in [S], e reduce this value of c to 3 by a different method In fact, [S, Theorem, 3] hold ith N > ep3 log, and [S, Theorem, ] hold ith N > ep8365 log Also, e improve the range of y, from y ep + ɛ log log log in [LP, Theorem ], to y > ep3 log Introduction We use p, q to denote prime numbers, and use c i to denote an absolute positive constants Let a be an integer Denote by l a p the multiplicative order of a modulo prime p Artin s Conjecture on Primitive Roots AC states that for non-square non-unit a, a is a primitive root modulo p for infinitely many primes p Thus, l a p = for infinitely many primes p Assuming the Generalized Riemann Hypothesis GRH for Dedekind zeta functions for Kummer etensions, C Hooley [H] shoed that l a p = p for positive proportion of primes p Thus, e epect that l a p is large, and close to p for large number of primes p On average, e epect that l a p/ behaves like a constant P J Stephens see [S, Theorem ] shoed that if N > epc log then for any positive constant A, here C is Stephens constant: N l a p p = CLi + O C = p p p 3 log A Although the value of the positive constant c is not eplicitly given in [S], but e see that c is at least e 9 see [S, Lemma 7] The optimal value of c along Stephens method is any positive number greater than e We replace by 3 by a different method Theorem If N > ep3 log, then for any positive constant D, N l a p p = CLi + O log D Similarly, e give an eplicit constant c in [S, Theorem ] Theorem If N > ep8365 log, then for any positive constant E, N l a p p CLi p< log E Keyords: Artin, Primitive Root, Average, AMS Subject Classification Code: A07,

2 Subsequently, e obtain an improvement of average results on P a = {p a is a primitive root modulo p} see [S0, Theorem, ]: If N > ep3 log, then for any positive constant D, 3 N P a = Aπ + O log D here A = p p is Artin s constant Also, the normal order result: If N > ep8365 log, then for any positive constant E, N P a Aπ log E Stephens also proved that the average number of prime divisors of a n b also asymptotic to CLi in [S, Theorem 3], and proved normal order result in [S, Theorem ] The number N is rather large compared to those of Theorem, N > log c in [S, Theorem 3], and N > log c in [S, Theorem ] respectively He mentioned that these could probably be improved by using the large sieve inequality as in Theorem, Hoever, he did not carry out the improvement in [S] Here, e state the improvement and prove them Theorem 3 If N > ep3 log, then for any positive constant D, 5 N = CLi + O log D b N p a n b for some n Theorem If N > ep8365 log, then for any positive constant E, 6 N b N p a n b for some n CLi log E In [C], Carmichael s lambda function λn is defined by the eponent of the group Z/nZ We say that a is a λ-primitive root modulo n if the order l a n of a modulo n is eactly λn Folloing the definitions and notations in [LP], Rn = #{a Z/nZ : a is a λ-primitive root modulo n}, N a = #{n : a is a λ-primitive root modulo n} S Li [L] proved that for y eplog 3/, 7 N a y a y n Rn n This as further improved by S Li and C Pomerance [LP], here they obtained 7 ith range of y: 8 y ep + ɛ log log log We prove 7 ith a ider range of y: Theorem 5 If y > ep3 log, then there eists a positive constant c such that 9 N a = Rn y n + O ep c log a y n

3 Proof of Theorems Proof of Theorem and We begin ith the folloing elementary lemma see [B]: Lemma Let r and define τ r a to be the number of ays to rite a as an ordered product of r positive integers If N, then e have 0 τ r a r! Nlog N + r r The proof is by induction Corollary Let c > 0 If N and r c log N, then + cr τ r a N log r N r! We define τ ra to be the number of ays of riting a as ordered product of r positive integers, each of hich does not eceed N Lemma We have Proof r τ ra = τ ra r τ r a r a,,a r N = τ ra τ ra a r r a,,a r N r = τ r a τ ra τ ra r Corollary Let c > 0 If N and r c log N, then + c 3 τ ra r r N log N r r! r We follo the notations in [S], and give a critical upper estimate of the folloing quantities: Lemma 3 Let S = and 5 S 0 = q p q χmod p χmod pq ordχ ordχ χa χa The sum denotes the sum over non-principal primitive characters Then by Hölder inequality and the large sieve inequality, 6 S r r + N r τ ra r

4 and 7 S 0 r r + N τ ra r Corollary 3 By taking N r < N r, e have for r c log N, 8 S N + + c r r N log r 3 N = N + + c r r r! r! By taking N r < N r, e have for r c log N, 9 S 0 N + + c r r N log r N = N c r r r! r! By [S0], e may assume that N = epk log Then N r the folloing: 0 log N = K log, log log N = log K + log log, r < log log N = log r K r log r N log r N = epok = log O and e have By Stirling s formula, e have S log O N ep log N + r log + c r logr! + log N ep K + K log + c K log + K + log K K + o From r c log N, e have K 3 S N ep ck Taking c =, e have K log K + K log + K K log + K + log K K + o If K 3, then e see that f K = K + K log + K K log + K + log K K < 0 We finally obtain that Lemma If N > ep3 log, then there is a positive constant c such that S N ep c log log log N No, e deal ith S 0 Let N = epk log By [S0], e may assume that K 6 log log Then = epok = log O and e have the folloing: N r 5 log N = K log, 6 log log N = log K + log log, 7 r < log log N = log r K

5 By Stirling s formula, e have S 0 log O N ep log N + r log + c r logr! + log N ep K + K log + c K log + K + log K K + o From r c log N, e have K 8 S 0 N ep ck Taking c =, e have K log K + K log + K K log + K + log K K + o If K 8365, then e see that f K = K + K log + K K log + K + log K K < 0 Therefore, e obtain that Lemma 5 If N > ep8365 log, then there is a positive constant c 3 such that 9 S 0 N ep c 3 log Denote by S 0 the folloing character sum: 30 S0 = q p q here Σ is over non-principal characters Then χmod pq 3 S0 N ep ordχ χa c 3 log log log N For the estimate for S 0, note that S 0 S 0 + S Proof of Theorem As in [S, Theorem ], e define a character sum c r χ here χ is a Dirichlet character modulo p: For r p, define 3 c r χ = χa p Then e have S 3 = N = N = N l ap= = N χmod p c χ p = p + O = CLi + O log A χa N + O + O N p a<p l ap= r + Olog log + O N log + O ep c log χmod p c χχa + O N S log log log N χmod p c χ χa by Lemma and [S, Lemma ] This completes the proof, since the second error term is dominated by the first

6 Proof of Theorem As in [S, Theorem ], let 33 S 5 = N l a p p CLi, then 3 S 5 = N Let q p q 35 S 6 = N Stephens decomposed S 6 into three parts: S 6 = N q p q = S 7 + S 8 + S 9 t q l a pl a q p q C Li + O log E t q p q χ mod p l a pl a q p q c χ χ mod q c t χ χ χ a here S 7, S 8, S 9 are the contributions to S 6 hen both χ, χ are principal, hen one of χ, χ is principal, and hen neither χ, χ is principal, respectively Then 36 S 7 = C Li + O log E + O N log, 37 S 8 log E, and 38 S 9 S 0 log log log N Then e have by Lemma 5, S 5 C Li C Li + O log E + O N log + O ep c 5 log Since the last to error terms are dominated by log E, this completes the proof of Theorem Proof of Theorem 3 and

7 Proof of Theorem 3 As in [S, Theorem 3], e rite the sum as N = N = N b N b N b N l b p l ap = N b N = N b N l b p l b p l b p l ap= χmod p c χ + O N b N = N b N l b p p χmod p p + O N log log τ p χa N + O + l b p N p c χ χa l ap= + O + Olog log N log χmod p ordχ χa To treat the last error term, let c be the positive constant in Lemma Choose any positive constant c 6 smaller than c, and split the sum into to parts: 39 N τ p ordχ χa = Σ + Σ, χmod p here Σ is the sum over p s ith τ p < epc 6 log, and Σ is the sum over remaining p s Then by Lemma, Σ N epc 6 log N ep c log ep c 7 log By an elementary estimate n τ n 3 log 7, Σ N Thus, e obtain 0 N b N l b p l ap τ epc 6 log τ p 3 epc 6 log ep c 7 log = N b N l b p τ p N p + O ep c 7 log

8 No, e treat the first sum on the right side Again, by riting it ith character sums, N = N p p b N here l b p = N = N = E = N We split the sum as before, E N log log p p p p t t τ = N log log τ 3 p = Σ 3 + Σ, t t b N l b p= t c t χ χmod p b N t t p t p χmod p c t χ b N χmod p χmod p χb N + O + O N + E p + O + Olog log + E, N log χb ordχ χb b N ordχ χb here Σ 3 is over p s ith τ 3 p < epc 6 log, and Σ is over remaining p s By the same argument, e have Σ 3 + Σ ep c 7 log b N Therefore, 3 N b N l b p l ap = p t t p + O ep c 7 log By the elementary identity d n d = n, e have N b N l b p l ap = p + O ep c 7 log Then Theorem 3 follos by [S, Lemma ]

9 Proof of Theorem Using the same argument as for Theorem e deduce that N b N p a n b for some n = N = N CLi b N q l b p l ap l b q l aq = N q u q χ,χ mod p t s q χ 3,χ mod q u C Li + O log E b N p a n b for some n q q a m b for some m C Li + O log E c χ c t χ c u χ 3 c us χ C Li + O log E χ χ 3 a b N χ χ b We prove that if any one of χ i, i =,, 3, is non-principal, then they contribute O / log E In this case, either χ χ 3 or χ χ is non-principal Suppose that χ χ 3 is non-principal Then, the contribution is N log log ordχ ordχ ordχ 3 ordχ χ χ 3 a q N log log N log log = E + E, q t t u q s q u u q s q u χ,χ mod p χ 3,χ mod q χ χ 3 is nonprincipal χ mod p χ 3 mod q χ χ 3 is nonprincipal χ mod p χ mod q τ 3 p τ 3 q τ p τ q q ordχ ordχ 3 χ mod p χ 3 mod q χ χ 3 is nonprincipal ordχ ordχ ordχ ordχ 3 χ χ 3 a χ χ 3 a here E is the sum over p, q s ith τ 3 p τ 3 q τ p τ q < epc 8 log, and E is the sum over remaining p, q s Here, e let c 8 be a positive number smaller than c 3 in Lemma 5 By Lemma 5, For E, e have E q E ep c 9 log τ 3 p τ 3 q τ p τ q τ 3p τ 3 q τ p τ q epc 8 log τ 3 p τ p 3 τ 3 q τ q 3 ep c 8 log q ep c 9 log The case hen χ χ is non-principal, is treated similarly and it also contributes ep c 9 log

10 No, e find that the main contribution is hen every character χ,, χ is principal In fact, N t q u q us N N N + O + O + O p p q q p q q Therefore, t = N = u q s q u q u q q u q = C Li + O log E 5 N b N q u p uq N + ON + O q u p uq + O N log p a n b for some n This completes the proof of Theorem CLi + O N p + O log log log N = C Li C Li + O log E q 3 Proof of Theorem 5 Folloing the definitions in [LP], then q n = #{cyclic factors C q v 6 Rn = n q n Let radm denote the largest square-free divisor of m Let En = {a Z/nZ : a λn radn in Z/nZ : q v λn}, q qn mod n}, and e say that χ is elementary character if χ is trivial on En For each square free h n, let ρ n h be the number of elementary characters mod n of order h Then ρ n h = q hq qn For a character χ mod n, let cχ = n here the sum is over λ-primitive roots in [, n] Then here cχ = { b cχ cχ, ρ nordχ For the proof of above, see [LP, Proposition ] χb, if χ is elementary, 0 otherise

11 Let Xn be the set of non-principal elementary characters mod n In [L], it is shon that 7 N a = y Rn + B, y + O log, n a y n here 8 B, y = cχ n χ Xn a y Folloing the proof in [LP], 9 B, y d µd S d, here 50 S d y d ep log 3 log log χa We use this hen d is large Let χ 0,n be the principal character modulo n For positive integer k and reals, z, define F k, z = cχχ 0,km m χa, and S, z = k radm k χmod k T, z = k F k, z, a z F k, z = T, z + T u, zdu, k u S d S d, y d We ant to estimate S d using the estimate of T, z Because of the integral in S, z, e need an estimate of T u, z for u First, assume that z 3 By applying Pólya-Vinogradov inequality, Li and Pomerance shoed in [LP, Lemma 5] that 5 T, z 3 ep 3 log log log z 3 ep Suppose no that > z 3 By Hölder inequality ith r = log log z, 5 T, z r A r B, here 53 A = m k radm k and 5 B = m k radm k χmod k χa a z χmod k r 3 cχχ 0,km r r, = k k k Then by 0 cχχ 0,km, log 55 A ep 3 log log χmod k log log log χa a z r

12 By large sieve inequality, 56 B + z r a z r τ ra If > z 3, then r > z 3 6 By the method in Lemma, e have Lemma 6 If z > ep8 log, then log 57 T, z z 3 6 ep f ɛ Lemma 7 If ep39906 log < z ep6 log, then log 58 T, z z 3 ep f ɛ Therefore, by S, z = T, z + u T u, zdu, Lemma 8 If z > ep8 log, then log 59 S, z z 3 6 ep f ɛ Lemma 9 If ep39906 log < z ep6 log, then log 60 S, z z 3 ep f ɛ + log z 7 8 Suppose that y > ep log If y d > ep 8 log d, then by Lemma 8, S d, y d y 6 ep log d d3 f d + ɛ y log ep f d 9 6 y ɛ y log ep 3 d f ɛ Since f < 0, the contribution of these d s is y ep c log is If y d ep 8 log d, then clearly d ep 00 log Then by 50, the contribution of these d s y ep c log Therefore, e obtain the folloing eaker version of Theorem 5: Theorem If y > ep log, then 6 N a = Rn y n + O ep c 3 log a y n No, assume that ep3 log < y ep6 log, then for large and for y d ep log, ep 6 log ep 6 log y d d d ep log > ep log d Thus by Lemma 9, 6 S d, y d d y log ep f d ɛ + y 7 d log 8 d

13 We sum this for y d ep log By f < 0, the contribution is log y ep f y log + ɛ + y 8 y ep c log If y d < ep log, then d > ep log By 50, the contribution of these d s is y ep c 5 log This completes the proof of Theorem 5 3 Some Numerical Data and Calculus Remarks The numerical values 39906, 39907, and 3 are positive numbers greater than the unique solution to the equation f K = 0 The values of f K for these numbers are negative Thus, 3 in Theorem, 3, 5 can be replaced by any positive number greater than the unique solution to f K = 0, hich is numerically Similarly, can be replaced by any positive number greater than the unique solution to 3 6 K + f K + K = 0, hich is numerically Also, 8365 in Theorem, can be replaced by any positive number greater than The function f K + K hich can be simplified as K log K + +, is a decreasing function for K > 0 and it converges to 0 as K The author used Wolfram Alpha for numerical calculations Further Developments The author thinks that the corresponding normal order result for Theorem 5 could probably be done such as: If y > ep8365 log, then 63 N a Rn ep c 6 log y n a y n The author also thinks that, in Theorem,, 3 and, if e replace CLi by, then log D, log E in the error terms may be replaced by ep c 7 log References [B] O Bordellės, Eplicit Upper Bounds for the Average Order of d nm and Application to Class Number, Journal of Inequalities in Pure and Applied Mathematics, Volume 3, Issue 3, Article 38, 00 [C] R D Carmichael, The Theory of Numbers, Wiley Ne York, 9 [H] C Hooley, On Artin s Conjecture, Journal für die Reine und Angeandte Mathematik, Volume 5, 967 pp 09-0 [L] S Li, An Improvement of Artin s Conjecture on Average for Composite Moduli, Mathematika 5 00, pp [LP] S Li, C Pomerance, The Artin-Carmichael Primitive Root Problem on Average, Mathematika 55009, pp 3-8 [S0] P J Stephens, An Average Result for Artin s Conjecture, Mathematika, Volume 6, Issue, December 969, p [S] P J Stephens, Prime Divisors of Second Order Linear Recurrences II, Journal of Number Theory, Volume 8, Issue 3, August 976