GENERALIZING THE TITCHMARSH DIVISOR PROBLEM

Transcription

1 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM ADAM TYLER FELIX Abstract Let a be a natural number different from 0 In 963, Linni roved the following unconditional result about the Titchmarsh divisor roblem log log d a = c log where c is a constant deendent on a Titchmarsh roved the above result assuming GRH for Dirichlet L-functions in 93 We establish the following asymtotic relation: a mod a d = C log where C is a constant deendent on and a and the imlied constant is deendent on We also aly it a question related to Artin s conjecture for rimitive roots Introduction Let dn denote the number of ositive divisors of n N Let denote a rime number Then, the Titchmarsh divisor roblem [2, 93] is concerned with the following summation: d a where a is a fied non-zero integer The above summation was first investigated by Titchmarsh [5], who roved the following theorem: Theorem Titchmarsh Suose the Generalized Riemann Hyothesis holds for Dirichlet L-functions Then d a = ζ2ζ3 log log ζ6 2 + log as a< a In 96, Linni [0] roved the above asymtotic formula with his disersion method, thereby eliminating the use of the generalized Riemann hyothesis from the above theorem 99 Mathematics Subject Classification N37, N36, N3 Key words and hrases Titchmarsh divisor roblem, rimes in arithmetic rogression, Artin s conjecture Research of the first author suorted by an NSERC PGS-D scholarshi

2 2 ADAM TYLER FELIX Rodriquez [4] and Halberstam [7] indeendently showed that the above can be roven unconditionally using the Bombieri-Vinogradov Theorem In fact, Fouvry [6, Corollaire 2], and Bombieri, Friedlander and Iwaniec [, Corollary ] have shown for any A >, 2 d a = c + c li log A where c and c are effectively comutable constants deendent on a and the imlied constant deends only on a and A, and li is the usual logarithmic integral Notation The letter will denote a rime number The letter will denote a ositive integer The function dn will denote the number of ositive divisors of n N The Euler totient function, which will be denoted by ϕn with n N, is the number of corime residue classes in Z/nZ Let a, N with gcda, = and R with Then, by π;, a denote by the number #{ : a mod }, and by ψ;, a we denote by the summation 3 ψ;, a := n n a mod Λn where Λn = log if n = α for some rime and α N and Λn = 0 otherwise That is, Λ is the von Mangoldt function For a, define f a = min{d N : a d mod } and for a, define f a = For a, define i a = and for a, define i f a a = 0 We call f a the order of a modulo and i a the inde of a modulo The logarithmic integral, denoted by li, is the integral 4 li := 2 dt log t Let f : R C and g : R R 0 We will write f = Og if there is a constant C such that f Cg for all R Equivalently, we will write f g for the same relation We will write f g if f g f Here f has codomain R 0 We will write f g if 5 lim f g = We will write f = og if 6 lim f g = 0 2 Statement of Results We wish to consider 7 d mod where N is fied We will also consider the above summation with d / relaced by d a/ as ranges over with a mod However, the case a = has

3 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 3 alications to Artin s conjecture for rimitive roots In 2, we will rove the following theorems: Theorem 2 For any N, > be an integer, and let a Z such that gcda, =, and A > 0 We have the following results uniformly in log A+ : 8 ϕd = c log log d gcda,d= where the O-constant is deendent only on a, and a d = c log + c 9 log where 0 a mod c := c a = w gcdw,a= = a + = ζ2ζ3 ζ6 µ 2 w gcdw, wϕw a a log A and the first O-constant is absolute and the second O-constant is deendent only on a and A In 3, we will rove Lemma Let a = Let c := c be defined as above Then, we have µc = 2 and Theorem 3 We have 2 as ϕ π;, = log These above theorems will then give an alication to a roblem related to Artin s conjecture for rimitive roots:

4 4 ADAM TYLER FELIX Theorem 4 Let y be a function of such that 3 and y a y f a Theorem 5 Let y be a function of such that 4 where y a y 5 c = ζ2ζ3 ζ6 where γ is Euler s constant: 6 γ := lim Let 2 δn := So, we have log = oy Then, y = log log log log = oy Then, ϕf a = ζ2ζ3 log + c log log ζ6 2 rime n log γ n log 2 Proof of Theorem 2 22 dn = 2 d n d< n Then, 23 Now, 24 a mod a d = 2 { if n Z 0 otherwise a mod = 2 d 2 + δn d a d a + a mod y a δ π; d, a δn n δn = #{n : n = m 2 for some m N} n = #{m 2 }

5 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 5 So, we have 25 a mod a d = 2 d 2 π; d, a However, since π; d, a if gcda, d >, we have 26 d 2 Hence, we have 27 d 2 π; d, a = d 2 gcdd,a= = li π; d, a = li = li π; d, a li ϕd + li ϕd d 2 gcdd,a= d 2 gcdd,a= d 2 gcdd,a= ϕd ϕd ϕd d 2 gcdd,a= d 2 gcdd,a=, log A d 2 gcdd,a π; d, a li ϕd π; d, a li ϕd where we have used [, Theorem 9] which states the following: choose ε > 0 If we assume < 0 ε which is true since, in our case, < log A+, then, there eists C := CA such that for any Q /log C we have 28 ψ; nm, a m ϕqr n Q log A gcda,m= gcdn,a= where the imlied constant deends on at most ε, a, and A, and C is deendent only on A We note that > / as < log A+/2 < /log C Therefore, by log B artial summation and concerning ourselves only with m =, which satisfies gcda, = by

6 6 ADAM TYLER FELIX hyothesis, in the above summation, we have 29 d 2 π; d, a = li d 2 gcdd,a= ϕd log A So, we just need to deal with d gcdd,a= ϕd We will evaluate this summation by using [2, Eercises 63 and 64] as a guide We have 20 d gcdd,a= d ϕd = d w d gcdd,a= = w µ 2 w ϕw = w gcdw,a= µ 2 w ϕw = µ 2 w ϕw w d w gcdw, d gcdd,a= µ 2 w ϕw = d gcdw, w gcdd,a= d w d gcdd,a= w w gcd gcdw,,a= µ 2 w ϕw d w gcdw, d gcdd,a= w since gcda, = imlies gcd, a w = gcdw, a, and gcd, a = imlies gcdw, gcdw, w gcd d, a = gcdd, a Also, since mutatis mutandis the roof of [, Eercise 58] gcdw, 2 d y gcdd,a= = ϕa a y da,

7 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 7 we have 22 However, 23 So 24 d gcdd,a= d gcdd,a= We notice that 25 d ϕd = w> w gcdw,a= µ 2 w ϕw = ϕa a = ϕa a µ 2 w gcdw, wϕw d ϕd = ϕa a w gcdw,a= w gcdw,a= ϕa a gcdw, w µ 2 w gcdw, wϕw µ 2 w gcdw, wϕw da log w> w gcdw,a= log = ϕa a w gcdw,a= w gcdw,a= µ 2 w wϕw µ 2 w gcdw, wϕw µ 2 w gcdw, wϕw µ 2 w gcdw, wϕw which converges absolutely for fied Define 26 c := c a := ϕa a Thus, we have 27 d gcdd,a= w gcdw,a= w da da w w> ϕw µ 2 w gcdw, wϕw log 2 w> log µ 2 w wϕw, µ 2 w gcdw, wϕw d ϕd = c log µ 2 w gcdw, wϕw

8 8 ADAM TYLER FELIX Note that 28 d gcdd,a= d gcdd,a= However, by artial summation, we have d dϕd = 29 d gcdd,a= ϕd = d d ϕd + log = c + = c log log u 2 d dϕd d u gcdd,a= d ϕd du c u du logu u 2 du since fw := gcdw, is a multilicative function imlies that we have c a µ 2 w gcdw, = 2 + wϕw 2 + w = d µ 2 d ϕd d ϕd log So, we have 22 Thus, 222 Thus, 223 d 2 d gcdd,a= π; d, a = li a mod ϕd = c log log = c log d 2 gcdd,a= ϕd log A = c 2 c log li + c log 2 2 log 2 log = c log + 2 c log log A a d = c log + c log log log A log A

9 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 9 So we just need to show that the Euler roducts are true To see this note that fw := gcdw, is a multilicative function Therefore, c a = ϕa µ 2 w gcdw, = µ n gcd n, a wϕw n ϕ n w a a n 0 gcdw,a= = gcd, + a a = + + a a 224 = ζ2ζ3 ζ6 a Therefore, Theorem 2 holds We note that we obtain an imrovement in this theorem by using Fiorilli s wor [4, Theorem 34] on etending Bombieri, Friedlander, and Iwaniec s wor [] This will give us imroved results in the net section 3 Alication to a Generalization of Artin s Conjecture Recall for a, the order of a modulo is 3 f a = min{d N : a d mod }, and for convenience, f a = for a We want to consider 32 f a = i a We note that if the above summation is /4, then there are infinitely many rimes for which a is a rimitive root see Murty and Srinivasan [2] We will show that, on average, the above summation is log Let us consider the following summation: 33 y a y f a We will also see that this summation is related to the Titchmarsh divisor roblem see [2, 93]: 34 d a = c + c li log A

10 0 ADAM TYLER FELIX for any A > 0, where c and c are constants, and the imlied constant is deendent only on a and A More recisely, this is related to the summation d mod discussed in the revious section Since f = for any Z and y = y+ y+, where is the smallest integer greater than R, we have f a y a f a a + + = y + 35 f a a since f a = f a+ by definition Therefore, since f a y + 36 a y = y + a y+ + a y+ a f a f a y+ = y+, we have f a a f a Similarly, using [] the greatest integer smaller than and truncating the above summations to a [y + /] instead of etending them to a y + /, we have 37 f a y a y + f a a f a a In articular, 38 a y a f a = y + = y a a f a a f a f a a f a Let us consider the error term first We note that the number of elements of Z/Z that have order is ϕ rovided and 0 otherwise So, we have 39 f a = #{a Z/Z : f a = } = ϕ

11 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM Therefore, 30 a f a = ϕ d by the Titchmarsh divisor roblem [2, 93] Let us now consider the main term We first note that we have 3 a f a = ϕ mod since 32 a f a = ϕ By artial summation, we have 33 mod = π;, + πu;, u 2 du Therefore, 34 a f a = ϕ π;, + πu;, u 2 du We will evaluate each of these sums searately The first summation is dealt with as follows: 35 ϕ π;, π;, = d by the Titchmarsh divisor roblem [2, 93] In order to evaluate the second summation notice that 36 ϕ πu;, u 2 du = 2 u 2 u So, in order to evaluate our desired sum, we need to evaluate 37 ϕ π;, ϕ πu;, du

12 2 ADAM TYLER FELIX We have 38 ϕ π;, = = = µ m µm m mod µ log A+2 + = m d mod log A+2 < µm m d µ mod mod m : m d Let us consider the second summation above We have 39 log A+2 < µ mod d log A+2 < log A+2 < log A+ log A mod log A+2 < d dn n /log A+2

13 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 3 For the first summation, we have µ d log A by Theorem 2 Also, 32 µc 2 mod µ c log + = c log log A+ log A+2 µc = + c log 2 log 2 log A+2 log A+2 log A+ log A+2 µc = + c log 2 log 2 log A+2 log A+2 log A µc 2 µc 2 = > µ µ 2 w gcdw, = µ µ 2 w gcdw, 2 wϕw 2 wϕw = w= > w= = + µ 2 + = µ > = µ ϕ > = =

14 4 ADAM TYLER FELIX This roves Lemma As c log, we have 322 So, c log 2 log A+2 ϕ π;, = log A+2 = log log as A > 0 can be chosen to be arbitrarily large Therefore, Theorem 3 holds It can now be shown that 325 = log log log y f a y assuming 326 We have 327 log a y = oy To see this, we need to evaluate 2 u 2 u 2 u 2 u ϕ πu;, du = 2 ϕ πu;, du u u u 2 log u = log log log by Theorem 3 This roves Theorem 4 since summation becomes 328 = log log log y f a a y log = oy forces y y du = olog and so our To see that Theorem 5 holds, note that all of the revious error terms and justifications wor for this case as well To see this, consider the following: since f = for any Z and y = y+ y+, we have ϕf a y a ϕf a a + + ϕf a y+ + a y+ = y ϕf a a

15 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 5 y+ since f a = f a+ by definition Therefore, since ϕf a y a y + = y + a 330 Similarly, we have 33 In articular, 332 Since 333 we have 334 a y a a y a ϕf a y + a ϕf a = y + = y ϕf a = a a a a = y+ a ϕf a, we have ϕf a ϕf a ϕf a ϕf a ϕf a ϕf a a #{ a : f a = } ϕ ϕf a = by the Titchmarsh divisor roblem Similarly, 335 ϕf a a = π;, + d a ϕf a ϕf a = = d, πu;, u 2 du The first summation is bounded by using the Titchmarsh Divisor roblem as before The integral becomes πu;, 336 du = πu;, du u 2 2 u 2 u

16 6 ADAM TYLER FELIX However, this inner summation becomes 337 Hence, u πu;, = u d = ζ2ζ3 ζ6 u u + c log u u log u πu;, u 2 du = ζ2ζ3 ζ6 log + c log log Therefore, Theorem 5 holds Remars We note that in both of Theorems 4 and 5, there is a log log term We also have 339 f a = i a We could aly artial summation the right-hand side but an estimate on the summation i a would be needed Currently, if we assume GRH for the Dedeind zeta functions of Qζ n, a /n as n ranges over N, we get lower bounds of the form 340 i a see [3, Chater 7, ] Unconditionally, we have 34 i a log log log To see this let π d := #{ : d i a } Then, by [8, Page 23], d i a if and only if slits comletely in Qζ d, a /d where ζ d is a rimitive d th root of unity So, by the unconditional effective Chebotarev density theorem [9, Theorem 4] or [3, Page 376], we have for any

17 GENERALIZING THE TITCHMARSH DIVISOR PROBLEM 7 A > 342 i a = ϕdπ d ϕdπ d d d log /7 = li ϕd [K d : Q] log A d log /7 li d by [6, Proosition 4] log A d log /7 log log log The belief [5, Conjecture a] is that we have 343 i a c a where c a is a ositive constant deendent on a Theorem 4 indicates that we may have 344 i a = c a log We note that in Theorem 4 and 5, there is a log log term In fact, in Theorem 5, this term contributes to the summation This suggests that we may have 345 i a = c a + Ω log As mentioned in the revious section, Fiorilli [4, Theorem 34] allow us to imrove Theorem 4 by relacing Olog log y by c log log y where c is a constant We relegate further analysis and comutation of this summation to future research Acnowledgements Some ortions of this wor were art of the doctoral thesis of the author [3] The author would lie Dr M Ram Murty, his suervisor, for comments on revious versions of this aer and guidance throughout the author s graduate school years at Queen s University The author wishes to than Dr Amir Abary and the reviewer for comments on revious versions of this aer The author also wishes to than Daniel Fiorilli for his comments and his rerint [4] References Enrico Bombieri, John Friedlander, and Henry Iwaniec, Primes in arithmetic rogressions to large moduli, Acta Math , Alina Carmen Cojocaru and M Ram Murty, An Introduction to Sieve Methods and their Alications, Cambridge University Press, New Yor, Adam Tyler Feli, Variations on Artin s rimitive root conjecture, PhD thesis, Queen s University, Kingston, Ontario, 20

18 8 ADAM TYLER FELIX 4 D Fiorilli, On a theorem of Bombieri, Friedlander and Iwaniec, rerint: arxiv:080439[mathnt] 20, 5 5 O M Fomeno, Class number of indefinite binary quadratic forms and the residual indices of integers modulo, J of Math Sci , no 6, Étienne Fouvry, Sur le roblème des diviseurs de titchmarsh, J Reine Agnew Math , Heini Halberstam, Footnote to the Titchmarsh-Linni divisor roblem, Proc Amer Math Soc 8 967, Christoher Hooley, On Artin s conjecture, J Reine Angew Math , Jeffrey Lagarias and Andrew Odlyzo, Effective versions of the Chebotarev density theorem, Algebraic Number Fields New Yor Albrecht Frohlich, ed, Academic Press Inc, 977, Ju V Linni, The disersion method in binary additive roblems, The disersion method in binary additive roblems Leningrad 96, Transl Math monograhs, vol 4, Amer Math Soc, Providence, Rhode Island, 963 M Ram Murty, Problems in Analytic Number Theory, second ed, Graduate Tets in Mathematics, no 206, Sringer, New Yor, M Ram Murty and Seshadri Srinivasan, Some remars on Artin s conjecture, Canad Math Bull , no, Francesco Paalardi, On Hooley s Theorem with weights, Rend Sem Mat Univ Pol Torino , G Rodriquez, Sul roblema dei divisori di Titchmarsh, Boll Un Mat Ital , no 3, E C Titchmarsh, A divisor roblem, Rend di Palermo 54 93, Samuel S Wagstaff, Jr, Pseudorimes and a generalization of Artin s conjecture, Acta Arith 4 982, 4 50 Deartment of Mathematics, Queen s University, Kingston, Ontario, K7L 3N6, Canada address: feli@mastqueensuca