Higher rank Artin conjecture with weight and average results

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(1)Scuola Dottorale in Scienze Matematiche e Fisiche Dottorato di Ricerca in Matematica XXVII Ciclo. HIGHER RANK ARTIN CONJECTURE WITH WEIGHT AND AVERAGE RESULTS. Candidato: Lorenzo Menici. Relatore: Prof. Francesco Pappalardi Coordinatore: Prof. Luigi Chierchia. Anno Accademico 2015/2016.

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(3) Introduction Artin’s conjecture (AC) on primitive roots is one of the most important open problems in number theory. The conjecture states that for any given integer a 0, 1 which is not a square there exist infinitely many prime numbers p such that a is a primitive root modulo p. More precisely, if Na pxq : #tp ¤ x : Fp xa pmod pqyu, then as x Ñ 8 Na pxq Apaq Lipxq ,. where the constant Apaq is a rational multiple of the so-called Artin’s constant A. ¹. . 1. 1. ppp 1q. p. .. Many attempts have been made to prove this conjecture and, so far, the only proofs of the “classical” AC and of many of its “variations” (like the higher rank AC, the weighted AC and the AC on average, studied in this thesis) rely on the Generalized Riemann Hypothesis (GRH). In particular, the classical AC has been proved under GRH by Hooley [9], so that we have the following: Theorem (Hooley). Consider an integer a 0, 1 which is not a perfect square and let h be the largest integer such that a ah0 ; denote with d the discriminant of the quadratic extension ? Qp aq. Assuming GRH for the number field Qpζn , a1{n q for every squarefree n, then . Na pxq δa Lipxq where µp n q δa rQpζn, a1{nq : Qs n¥1 ¸. with. Oa. x log log x log2 x. ,. $ h ' &A . ' %. Aphq . pq. ± 1 ± 1 1 µp|d|q q|d q2 q|d q q1 Aphq 2. |. q-h. qh. ¹ p-h. . 1. 1. ppp 1q i. ¹. |. ph. . 1. 1. p1. .. if d 1 p mod4q , if d 1 p mod4q ,.

(4) A possible generalization of the classical AC1 is the so-called “higher rank AC” or “r-rank AC”, meaning that we now deal with a finitely generated subgroup of the rationals Γ Q of rank r 8 (instead of the classical 1-rank conjecture): this subgroup will have the form Γ xg1 , . . . , gr y where pg1 . . . gr q is an r-tuple of multiplicatively independent rational numbers and its reduction Γp xg1 pmod pq, . . . , gr pmod pqy is well-defined for almost all prime numbers. In Chapter 1 we will see some results concerning the r-rank AC, where one is interested in those primes p for which the index ip : rFp : Γp s 1, together with the “quasi” r-rank AC, where one looks after those primes for which ip m for a certain natural number m dividing p 1: these results will be used to prove the original results of Chapters 2 and 3. Chapter 2 deals with what can be called “weighted r-rank AC”, meaning that we consider the sum ¸ f pip q (1). ¤. p x. where f pnq is a generic arithmetic function that “weights” the indices ip . This is the r-rank generalization of the work of Pappalardi [23], where the specific case Γ x2y was considered. Various theorems on the asymptotic behavior for x Ñ 8 of the sum (1) are presented, both unconditionally and under GRH, together with different applications. The original results of this Chapter appear in the submitted paper [16]. In Chapter 3 is presented a joint work with Cihan Pehlivan [17] on the r-rank AC on average. This work is the higher rank generalization of the original work of Stephens [31], where it was proved that, if T ¡ expp4plog x log log xq1{2 q, then ¸ ϕpp 1q 1 ¸ Na pxq T a¤T p1 p ¤x. . O. x plog xqD. . A Lipxq. O. x plog xqD. ,. where D is an arbitrary constant greater than 1; in the same paper it was also proved that, assuming T ¡ expp6plog x log log xq1{2 q, then. for any constant D1 xa1, , ar y, with ai. 1 ¸ tNapxq A Lipxqu2 T a¤T. 2. ! plogxxqD1. ,. ¡ 2. In Chapter 3 the analogous averages are studied, for the case Γ P Z for all i 1, . . . , r; the following unconditional theorems are proved: Theorem. Let T : mintTi : i 1, . . . , ru ¡ expp4plog x log log xq q and m ¤ plog xqD for an 1 2. arbitrary positive constant D. Then 1. T1 Tr. ¸. P ¤. Nxa1 , ,ar y,m pxq Cr,m Lipxq. ai Z 0 a1 T1. . O. x plog xqM. .. .. ¤. 0 ar Tr 1. For an exhaustive survey that discuss different generalizations of AC, see [20].. ii. ,.

(5) pq ¥ p q p q and M ¡ 1 is arbitrarily large. 1 Theorem. Let T ¡ expp6plog x log log xq 2 q and m ¤ plog xqD for an arbitrary positive constant where Cr,m :. °. µ n n 1 nm r ϕ nm. D. Then 1. T1 Tr. ¸. P ¤. Nxa1 , ,ar y,m pxq Cr,m Lipxq. ai Z 0 a1 T1. (2. 2. ! plogxxqM 1. .. .. 0 ar ¤Tr 1 where M ¡ 2 is arbitrarily large.. Finally, in Chapter 4 we expose the ongoing work that will be subject of a future joint paper with Francesco Pappalardi [18].. iii.

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(7) Notations We give a list of notations we will use in the thesis. We denote with Z the ring of integers and with Q, R, C respectively the field of rational, real and complex numbers. If not differently specifies, p, q and ` will always denote prime numbers. Fp is the finite field with p elements. Given an integer n 0, its p-adic valuation is vp pnq maxtk P N : pk | nu; hence, the p-adic valuation of a rational number g a{b is defined as vp pg q vp paq vp pbq. In the whole thesis, Γ Q is a finitely generated subgroup of the rationals with rank r: we can think about it as Γ xg1 , . . . , gr y with generators gi multiplicatively independent, that is g1e1 g2e2± grer 1 only if e1 e2 . . . er 0. We define SupppΓq tp : vppgq 0, g P Γu and σΓ pPSupppΓq p. The reduction of Γ modulo a prime p is well defined for every prime p - σΓ and it’s denoted as Γp xg1 pmod pq, . . . , gr pmod pqy. The order of Γp is indicated as |Γp | and the relative index is ip rFp : Γp s. For every positive integer n, we set Kn pΓq : Qpζn , Γ1{n q, the (Kummer) field generated by the n-th roots of the elements of Γ and by the n-th root of unity ζn e2πi{n ; we also indicate with kn pΓq : rKn pΓq : Qs the relative degree. For two real functions f pxq, g pxq, we write f pxq Opg pxqq (or equivalently f pxq ! g pxq) as x Ñ 8 if and only if there exists a constant C ¡ 0 and a real number x0 such that |f pxq| ¤ Cg pxq for all x ¥ x0 . As a consequence, Op1q stands for an arbitrary constant. We write f pxq opg pxqq if limxÑ8 f pxq{g pxq 0. The logarithmic integral Lipxq is defined as Lipxq :. »x 2. dt log t. and from the Prime Number Theorem we know that, asymptotically as x Ñ 8, Lipxq x{ log x π pxq, where π pxq : #tp ¤ xu is the prime counting function. Given a number field K (i.e. a finite-dimensional field extension of Q), we consider the set I of the non-zero ideals I of its ring of integers OK . The Dedekind zeta function of K is then defined as ¸ 1 ζ K ps q : pN pI qqs I PI v.

(8) where N pI q rOK : I s is the norm of the ideal I and s σ it is a complex number: it can be shown that ζK psq has an analytic continuation to a meromorphic function on Czt1u, having a single pole at s 1. The Generalized Riemann Hypothesis (GRH) is an unproven conjecture which says that if ζK psq 0 and 0 σ 1, then σ 1{2.. vi.

(9) Acknowledgements First of all, I want to thank my advisor, Francesco Pappalardi, for all the help, answers and patience he has generously offered to me during all the period of preparation of this thesis. Moreover, I would like to thank him for his moral support and understanding in my difficult times. I also would like to thank Cihan Pehlivan for his efforts in our joint work on the average of r-rank Artin’s conjecture. Finally, I want to thank my partner Angelita and my parents for their constant support during these years.. vii.

(10) viii.

(11) Contents Introduction. i. Notations. v. Acknowledgements. vii. 1 Higher rank Artin’s conjecture on primitive roots: 1.1 Artin’s conjecture on primitive roots . . . . . . . . . 1.1.1 The Chebotarev Density Theorem . . . . . . 1.2 r-rank Artin’s conjecture . . . . . . . . . . . . . . . . 1.3 r-rank quasi-Artin’s conjecture . . . . . . . . . . . .. an . . . . . . . .. overview . . . . . . . . . . . . . . . . . . . . . . . .. 2 r-rank Artin’s conjecture with weights 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ° 2.2 Main results for p¤x, p-σΓ f pip q . . . . . . . . . . . . . . . . . . 2.2.1 Conditional case . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Unconditional case . . . . . . . . . . . . . . . . . . . . . 2.3 Examples and applications . . . . . . . . . . . . . . . . . . . . . 2.3.1 f pnq multiplicative function . . . . . . . . . . . . . . . . 2.3.2 f pnq χB pnq characteristic function of a subset B N 2.3.3 Indices ip in arithmetic progression . . . . . . . . . . . . 2.3.4 Average order of Γp over primes . . . . . . . . . . . . . 3 Average r-rank Artin’s conjecture 3.1 Introduction . . . . . . . . . . . . . 3.2 Notations and conventions . . . . . 3.3 Lemmata . . . . . . . . . . . . . . 3.4 Proof of Theorem 1 . . . . . . . . . 3.5 Proof of Theorem 2 . . . . . . . . .. . . . . .. . . . . . ix. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. . . . . . . . . .. . . . . .. . . . .. 1 1 2 3 5. . . . . . . . . .. 7 7 9 9 11 14 15 20 23 25. . . . . .. 27 27 30 30 38 44.

(12) 4 Work in progress. 51. Bibliography. 55. x.

(13) Chapter 1. Higher rank Artin’s conjecture on primitive roots: an overview In this Chapter we present some results that will be used in Chapters 2 and 3.. 1.1. Artin’s conjecture on primitive roots. We describe in this Section the basic ideas behind the study of Artin’s conjecture on primitive roots; we first focus on the classical 1-rank case, then we generalize the problem for a generic (finite) rank r. In 1927 Artin conjectured that, given any number a P Qzt0, 1u which is not a perfect square, then there exist infinitely many prime numbers p for which a is a primitive root modulo p. Moreover, Artin estimated that if we denote Na pxq : #tp ¤ x : rFp : xa pmod pqys 1u, then Na pxq δa Lipxq ,. as x Ñ 8, where δa is a non-vanishing constant depending only on a. For a fixed prime number p, a rational number a such that vp paq 0 is a primitive root p1 modulo p if and only if a q 1 pmod pq for every prime q | pp 1q. In order to study the asymptotic behavior of Na pxq as x Ñ 8 we can think of fixing q in order to look after those p1 primes p 1 pmod q q such that a q 1 pmod pq and we do that for every fixed prime number q. Consequently, a is not a primitive root modulo p if the conditions p 1 pmod q q and p1 a q 1 pmod pq are simultaneously satisfied. But p 1 pmod q q if and only if the equation p1 xq 1 pmod pq has q distinct solutions and a q 1 pmod pq if and only if the equation p1 xq a pmod pq is solvable. Then, the primes p 1 pmod q q satisfying a q 1 pmod pq split 1.

(14) completely in the field Kq paq Qpζq , a1{q q; since the converse is also true, we have p 1 pmod q q ^ a. . p 1 q. 1 pmod pq ðñ p splits completely in Kq paq .. So if we indicate with Pp rKq paqs the probability that a prime p splits completely in Kq paq, a naive probabilistic approach to Artin’s conjecture should lead to the probability ¹. p1 PprKq paqsq. q. for a to be a primitive root modulo p. Unfortunately, for generic primes p, p1 , q, q 1 the events Pp rKq paqs and Pp1 rKq1 paqs are not in general independent and the previous reasoning must be modified, as was already known to Artin himself by the numerical results of the Lehmers [14].. 1.1.1. The Chebotarev Density Theorem. One of the main tools used in the study to Artin’s conjecture on primitive roots is the Chebotarev Density Theorem [3]. This powerful theorem can be stated as follows: Theorem 1.1.1 (Chebotarev). Let F be a number field and K {F a finite Galois extension with Galois group G GalpK {F q; given any conjugacy class C G, then the set of unramified primes p OF for which the Frobenius substitution σp has conjugacy class C has density |C |{|G|.. In our case, we are interested in those primes p P Z that split completely in Kq paq, so that the conjugacy class we are dealing with is the trivial one C tidu. From Chebotarev Density Theorem then #tp ¤ x : splits completely in Kq paqu . 1 rKq paq : Qs Lipxq ,. as x Ñ 8. For any squarefree n, we can consider the compositum Km paq of the fields Kq paq with q | m and we have Km paq Qpζm , a1{m q; since we are looking for those primes p that do not split in any Kq paq, by the inclusion-exclusion principle we are led to formulate Artin’s conjecture as follows: Conjecture 1.1.2 (Artin). Let a P Qzt0, 1u and h maxtn P N : a P Q n u, then Na pxq #tp ¤ x : rFp : xa pmod pqys 1u δa Lipxq ,. with µpmq δa rKmpaq : Qs m ¥1 ¸. $ h ' &A . ' %. pq. ± ± 1 µp|d|q q|d q1 2 q|d q 1q1 Aphq 2. |. qh. 2. q-h. xÑ8, if d 1 p mod4q , if d 1 p mod4q ,.

(15) . p p, hq 1 Aphq . ppp 1q p Now let ip : rFp : xa pmod pqys, Supppaq : tp : vp paq 0u and, for any squarefree m, πa px, mq : tp ¤ x : p R supppaq, m | ip u; we notice that, for any prime p R Supppaq, p 1 pmod mq ^ a 1 pmod pq ðñ m | ip .. and. ¹. p 1 m. The following theorem is effective version of Chebotarev Density Theorem due to Lagarias and Odlyzko [12]: Theorem 1.1.3. Assuming GRH for the Dedekind zeta function of Km paq, then πa px, mq . 1 rKmpaq : Qs Lipxq. ?. Op x logpmxqq .. (1.1). Unconditionally, there exists an absolute constant A such that, if m ¤ plog xq1{7 , then πa px, mq . 1 rKmpaq : Qs Lipxq. a. Opx exppA log x{mqq .. (1.2). Conjecture 1.1.2 was first proven by Hooley [9] assuming GRH for the Dedekind zeta function associated to the field Km paq for any m. The necessity of assuming GRH can be explained as follows: one can prove unconditionally that, if ξ pxq is a real-valued function that goes to infinity slowly enough, then the proportion of prime numbers p that do not split completely in Kq paq for any prime q ¤ ξ pxq is exactly δa as x Ñ 8, but one cannot prove the same for those primes q ¡ ξ pxq. The reason for that lies in the tail of the inclusion-exclusion: without GRH, the errors for all the prime numbers q ¡ ξ pxq due to unconditional version of Chebotarev Density Theorem (1.2) give a contribution that overwhelms the main term δa x{ log x in Artin’s conjecture. But, assuming GRH, the errors from (1.1) in the tail give an overall error which is opx{ log xq.. 1.2. r-rank Artin’s conjecture. One of the most impressive results on Artin’s conjecture is the unconditional theorem by Gupta and Ram Murty [7] and its successive refinement by Heath-Brown [10]: it is a 3-rank case whose main result can be stated as follows: Theorem (Gupta, Ram Murty, Heath-Brown). Let a1 , a2 and a3 be non-zero multiplicatively independent integers. Suppose that none of a1 , a2 , a3 , 3a1 a2 , 3a1 a3 , 3a2 a3 and a1 a2 a3 is a square. Then for at least one i P t1, 2, 3u we have Nai pxq " 3. x . log2 x.

(16) If we consider Γp , the reduction modulo p of a finitely generated subgroup of rationals Γ Q of rank r ¥ 1, the problem is now to study the asymptotic behavior of the quantity NΓ pxq : #tp ¤ x : rFp : Γp s 1u. The strategy is similar to the one used in the 1-rank case but, instead of the field Km paq previously discussed, we now need to take into account the field Km pΓq Qpζm , Γ1{m q. This problem has been studied by Pappalardi in [24] and few years later together with Cangelmi in [2]: Theorem 1.2.1 (Pappalardi). Let Γ Q be a finitely generated subgroup of rank r. Assuming GRH for the Dedekind zeta function of Qpζm , Γ1{m q, we have x NΓ pxq δΓ log x. . plog xqr. O. x 1 plog log xqr. ,. where the implied O-constant depends only on Γ and δΓ. . µ pn q rQpζm, Γ1{mq : Qs . m ¥1 ¸. The density δΓ was first computed in [24] for the case Γ xp1 , . . . , pr y with pi odd prime for every i 1, . . . , r and successively in the general case Γ Q in [2]. In order to perform density computations in the r-rank case, we need expressions for the degree kn pΓq : rKn pΓq : Qs. We know (see Lang [13, Theorem 8.1]) that the degree of the extension kn pΓq ϕpnq# tΓQpζn q n {Qpζn q n u .. (1.3). In particular, to express δΓ as an Euler product, we should exploit a useful formula for kn pΓq; we can investigate the properties of this extension degree using the results in [25, Lemma 1, Corollary 1]: Lemma 1.2.2. (Pappalardi) Let Γ Q , if α v2 pnq is the 2–adic valuation, then kn pΓq where. ϕpnq |Γpnq|. |Γrpnq|. ,. (1.4). Γpnq Γ Qn {Qn. and Moreover, if Γ Q , then. !. r pnq pΓ X Qpζn q Γ. r pn q m P N : m | σ Γ , m 2 Γ. . 2α. q Q {Q 2α. Q. 2α. . ). P Γp2αq, ∆pmq | n , ? where ∆pmq is the field discriminant of the real quadratic extension Qp mq. α 1. 4. 2α. (1.5).

(17) Different results in this thesis depend on equation (1.5), which holds only for r-rank positive subgroups of the rationals. To widen the validity of these results, the author, in collaboration with Francesco Pappalardi, is currently working on a generalization of equation (1.5) to the case of arbitrary finitely generated subgroups Γ Q , as will be briefly discussed in Chapter 4. Similarly to the 1-rank case, the key ingredient for the proof of Theorem 1.2.1 is the Chebotarev Density Theorem, which provides us with an asymptotic formula with error for πΓ px, nq : # tp ¤ x : p - σΓ , n|ip u. The following statement is obtained using the effective versions[30, Théorème 4] and [26, Lemma 4.1] in the conditional case, [12, Theorem 1.3–1.4] and [25, Lemma 4] in the unconditional case. Theorem 1.2.3 (Chebotarev Density Theorem). Let Γ Q be a finitely generated subgroup of rank r and n P N . suppose that the Generalized Riemann Hypothesis (GRH) holds for the Dedekind zeta function of Kn pΓq. Then, as x Ñ 8, πΓ px, nq . 1 lipxq kn pΓq. OΓ. ?x logpxnr 1q .. (1.6). Unconditionally, there exist constants c1 and c2 depending only on Γ such that, uniformly for n ¤ c1 as x Ñ 8,. 1.3. πΓ px, nq . . log x plog log xq2. 1 lipxq kn pΓq. 1{p3r. ,. . OΓ. q. 3. x. ec2. ?6 log x ?3 log log x. .. (1.7). r-rank quasi-Artin’s conjecture. For every m dividing p 1, let NΓ px, mq : #tp ¤ x : p - σΓ , ip. mu ;. (1.8). the asymptotic behavior of NΓ px, mq is the subject of study of the so-called r-rank quasi-Artin’s conjecture. Concerning the case r 1, the first result of this kind, due to Murata, appears in [22]: Theorem 1.3.1. (Murata) Let Γ xg y with g for every ¡ 0 we have. P Qzt1u not a perfect square. Assuming GRH,. Nxgy px, mq ρxgy,m lipxq where the implied constant depends only on . 5. . O. m x log log x log2 x. ,. (1.9).

(18) Recently in [26] Pappalardi and Susa proved the following: Theorem 1.3.2. (Pappalardi, Susa) Let Γ Q be finitely generated subgroup of rank r ¥ 2 and let m P N. Assume that the GRH holds for the fields of the form Qpζk , Γ1{k q with k P N. r 1 Then, for any ¡ 0 and for m ¤ x pr 1qp4r 2q , NΓ px, mq where. In particular, if Γ Q. . . ρΓ,m. ϕpmr. O. ρΓ,m :. 1 1. q logr x. lipxq,. µ pn q . k p Γ q mn n¥1 ¸. (1.10). tq P Q : q ¡ 0u and with the notation Γpkq Γ Qk {Qk , ρΓ,m. !. . ¹ 1 1 1 ϕpmq|Γpmq| p¡2 pp 1q|Γppq|. .. p-m. Notice that ρΓ,m is a rational multiple of µp n q Cr r ϕpnq n n¥1 ¸. . ¹ p. the so-called r-Artin’s constant.. 6. . 1. 1 pr pp 1q. ,. (1.11).

(19) Chapter 2. r-rank Artin’s conjecture with weights The results discussed in this Chapter appears in the work [16].. 2.1. Introduction. Given an arithmetic function f : N Ñ C, we study the asymptotic behavior of the sum ¸. ¤. f pi p q ,. xÑ8.. (2.1). p x p-σΓ. In the case f pnq χt1u pnq . #. 1 0. if n 1 , if n ¡ 1 ,. we obtain the generalization of the original Artin’s conjecture on primitive roots to the case of subgroups of rationals with finite rank r for which the main result is Theorem 1.2.1. The sum (2.1) is the r-rank generalization of the analogous sum considered in [23], where the author focused on the case Γ x2y. Notice that, from (1.8), the sum (2.1) can be rewritten as ¸. ¤. p x p-σΓ. Setting. f pip q . ¸. ¥. f pmq NΓ px, mq .. m 1. πΓ px, nq : # tp ¤ x : p - σΓ , n|ip u , 7. (2.2).

(20) we have shown in Chapter 1 that, when Γ xay, with a P Zzt0, 1u being not a perfect square, the intuition behind Hooley’s proof of Artin’s conjecture (under GRH) is based on the identity # tp ¤ x : p - σΓ , ip. 1u . ¸. ¥. µpnqπΓ px, nq,. (2.3). n 1. which is nothing but the inclusion-exclusion principle. The natural generalization of (2.3) is the identity ¸ ¸ f pi p q frpnqπΓ px, nq , (2.4). ¤. ¥. p x p-σΓ. n 1. °. °. where, by M¨ obius inversion, frpnq d|n µpn{dqf pdq and f pmq n|m frpnq. To prove the unconditional results of Section 2.2.2, we will provide in Lemma 2.2.3 the unconditional analogue of Theorems 1.3.1 and 1.3.2.. Remarks/assumptions. • The sum appearing in the right-hand side of equation (2.2) is finite: in fact, NΓ px, mq 0 if m ¥ x; • from the estimate (1.11), obviously ρΓ,m. ! 1{ϕpmq;. • if not conversely specified, p and q indicate prime numbers; • we will indicate pa, bq : gcdpa, bq; • in the whole paper we will suppose that ¸. ¥. |f pmq| ρΓ,m 8. (2.5). m 1. together with. | p q| 8 . k pΓ q m¥1 m ¸ fr m. (2.6). For the theorems we are going to prove, the results of Matthews [15] are vital and, in our particular case, they can be summarized through the following: Lemma 2.1.1. (Matthews) Let Γ Q be a finitely generated subgroup of rank r every t P R, t ¡ 1, we have #tp R supppΓq : |Γp | ¤ tu OΓ. 8. . t1 1{r log t. .. ¥ 2; then for.

(21) 2.2. Main results for. 2.2.1. °. p¤x, p-σΓ. f pi p q. Conditional case. The following theorems are r-rank generalizations of the conditional statements in [23, Theorem 2]: Theorem 2.2.1. Let Γ Q be a finitely generated subgroup of rank rand let f : N Ñ C be an arithmetic function such that maxm¤z t|f pmq|u ! plog z qC , for some positive real constant C; assuming that the GRH holds for the Dedekind zeta function of the field Qpζk , Γ1{k q for each k P N, then ¸. ¤. p x p-σΓ. where frpnq . °. |. ¸. f pip q . ¥. f pmqρΓ,m Lipxq . m 1. dnµ. p q Lipxq , k pΓq n¥1 n ¸ fr n. as x Ñ 8 ,. (2.7). pn{dqf pdq.. ¥ 2. Using the identity (2.2), we split our sum as ¸ ¸ f pmq NΓ px, mq f pmq NΓ px, mq f pmq NΓ px, mq .. Proof. We first consider the case r ¸. ¤. . ¥. y m z. m y. (2.8). m z. We first deal with the last sum in (2.8) and, in order to do that, we use the results of Lemma 1 2.1.1, under the choice z x r 1 plog xqB , with B ¥ C: ¸ f m NΓ x, m m¥z . p q p. ! plog xqC OΓ. . q ¤ max t|f pmq|u # m¤x. ¤. m y. f pmq NΓ px, mq . . ¸. ¥. m 1. ¸. ¥. p R supppΓq : |Γp | ¤. x) z. px{zq1 1{r opLipxqq . logpx{z q. Concerning the first sum in (2.8), we choose y Theorem 1.3.2 and get ¸. !. f pmqρΓ,m Lipxq f pmqρΓ,m Lipxq. . O. . r 1 x pr 1qp4r 2q. ¸. ¡. 1. so that we can make use of. f pmqρΓ,m Lipxq. m y. . O. Lipxq r ϕpm 1 q logr x. opLipxqq ,. m 1. °. where the last equality comes from the hypothesis that m¥1 |f pmq|ρΓ,m converges. Finally, the middle term in (2.8) is estimated using the conditional version of Chebotarev Density Theorem, 9.

(22) equation (1.6), as ¸ f m NΓ x, m y m z. p q p. . q ¤ plog zq O. ¸. C. . πΓ px, mq plog z q. y m z. . xplog xqC 1. ¸. 1. km pΓq. C. ¸. . . y m z. Lipxq km pΓq. O z plog xqC. 1. OΓ. ?x logpxmr 1q. ?x .. m z. From Hooley’s work [9], we know that if kn paq rQpζn , a1{n q : Qs is the degree of a Kummer extension for a fixed integer a, a 0, 1, then writing a bh for some integer b b0 b21 , with b0 squarefree integer and h maxtn P N : a bn u, we have kn paq . nϕpnq , δ pnqpn, hq. where δ pnq 1, 2 depends on the congruence class of a (mod 4); now, if g1 a{b with pa, bq 1, since Qppa{bq1{n q Qppabn1 q1{n q, the previous argument still works to give the lower bound km pΓq rQpζm , g1 1{m , . . . , gr 1{m q : Qs ¥ rQpζm , g1 1{m q : Qs ¥ for a fixed C. mϕpmq C. (2.9). P N. In the end, using Abel’s summation,. ¸ f m NΓ x, m y m z. p q p. The proof still works when r. . q O. x. plog xqC z. O z plog xqC. 1. ?x opLipxqq .. 1 if, instead of Theorem 1.3.2, we exploit Theorem 1.3.1.. Theorem 2.2.2. Let Γ Q be a finitely generated subgroup of rank r and let f : N Ñ C be an arithmetic function such that f pmq ¥ 0 for every m; assuming that the GRH holds for the Dedekind zeta function of the field Qpζk , Γ1{k q for each k P N, then ¸. ¤. p x p-σΓ. f pip q Á. ¸. ¥. f pmqρΓ,m Lipxq ,. where g pxq Á hpxq if for every ¡ 0 exists x such that |g pxq| ¥ p1 Proof. Suppose initially that r. (2.10). m 1. q|hpxq| whenever x ¡ x .. ¥ 2. The proof is easily adapted from the one of [23, Theorem 2.b] 10.

(23) xα with 0 α pr 1rqp4r1 ¸ ¸ ¸ f pmq NΓ px, mq f pip q f pmq NΓ px, mq ¥. and proceeds as follows: take y. ¤. m y. m 1. . . ¸. . f pmq ρΓ,m. ¤. O. m y. . ¸. ¥. f pmq ρΓ,m Lipxq . . ¸. ¥. ϕpmr ¸. ¡. 1 1. q logr x. Lipxq. f pmq ρΓ,m Lipxq O. . m y. m 1. . q , then by Theorem 1.3.2. ¤. ¥. p x p-σΓ. 2. ¸ f pmq 1 logr x m¤y ϕpmr 1 q. Lipxq. f pmq ρΓ,m op1q Lipxq ,. m 1. °. where we have used the convergence of the series m¥1 f pmq{ϕpmr 1 q, which is a consequence of the assumption (2.5) together with the bound |Γpmq| ¥ mr {∆r pΓq|, where ∆r pΓq is a computable constant which depends only on Γ (see [25], equation (7)). Also in this proof, we deal with the case r 1 using the results in Theorem 1.3.1 analogously to what we have done before with Theorem 1.3.2 when r ¥ 2.. 2.2.2. Unconditional case. Here follows an upper bound for NΓ px, mq without assuming GRH and, to to this, we need the unconditional statement in Theorem 1.2.3: Lemma 2.2.3. Let Γ Q be a finitely generated subgroup ofrank r ¥ 2; there exists a α x with α 3r1 3 , computable constant c1 depending only on Γ such that, if m ¤ c1 ploglog log xq2 then. 1 NΓ px, mq ¤ ρΓ,m Lipxq o Lipxq . (2.11) ϕpmr 1 q Proof. As a consequence of the inclusion-exclusion principle we can write NΓ px, mq . ¸. ¥. µpk q πΓ px, mk q .. k 1. For every y. P r1, xs, if P pyq denotes the product of the prime numbers less or equal than y then ¸ NΓ px, mq ¤ µpdq πΓ px, mdq . | pq. dP y. Given the computable constant c1 of Theorem 1.2.3, we choose y such that mP py q ¤ c1. . log x plog log xq2 11. 1 3r 3. ,. (2.12).

(24) so that we can apply the unconditional version of the Chebotarev Density Theorem: NΓ px, mq. ¤. ¸. | pq. µpdq. . dP y. . Lipxq kmd pΓq . . OΓ. . x exp c2 plog xq { plog log xq1{3.

(25) ¸ Li x. O. ρΓ,m Lipxq. pq k pΓq d¥y md. 1 6. . OΓ 2πpyq x exp. . . . c2plog xq1{6plog log xq1{3(2.13),. for a certain computable constant c2 . From [26, Corollary 4.3] we know that 2 r ∆ r pΓ q ¤ pΓq pmdqr ϕpmdq. 1 kmd. for a certain constant ∆r pΓq depending only on Γ. Through partial summation we obtain the following bound: ¸ 1 ¸ 1 1 1 ! ! . r 1 r r k ϕpm q d¥y d ϕpdq y ϕpmr 1 q d¥y md Choosing y log log log x, which satisfies the condition (2.12) as can be seen from [28, formula (3.15)], the two error terms in (2.13) become . O. Lipxq r y ϕpmr. 1. q. . OΓ. 2 p q x exp π y. . c2plog xq { plog log xq1{3 1 6. . o. . Lipxq ϕpmr 1 q. .. Theorem 2.2.4. Let Γ Q be a finitely generated subgroup of rank r ¥ 2 and let f : N Ñ C be an arithmetic function such that maxm¤x t|f pmq|u ! x1δ , for some real constant 0 δ 1; ° if the series n¥1 |f pnq|{ϕpnq converges, then ¸. ¤. f pi p q À. p x p-σΓ. ¸. ¥. f pmqρΓ,m Lipxq ,. where g pxq À hpxq if @ ¡ 0 exists x such that |g pxq| ¤ p1 Proof. We start splitting the (finite) sum ¸. ¤. m y. f pmqNΓ px, mq. as x Ñ 8 ,. m 1. °. ¸. . ¥. m 1f. pmqNΓpx, mq as. f pmqNΓ px, mq. y m z. q|hpxq| whenever x ¡ x .. ¸. ¥. m z. 12. f pmqNΓ px, mq .. (2.14).

(26) From Lemma 2.2.3, there exists a computable constant c1 such that, choosing y then ¸. ¤. ¸. f pmqNΓ px, mq ¤. m y. ¤. . . f pmq ρΓ,m. o. m y. ¸. . ¥. m 1. ¸. ¥. f pmqρΓ,m Lipxq f pmqρΓ,m Lipxq. 1 ϕpmr . 1. q. ¸. O. ¡. c1. . log x log log x. p. q2. 1{p3r. Lipxq . f pmqρΓ,m Lipxq. m y. ¸. f pm q o. ¤. . m y. opLipxqq .. Lipxq ϕpmr 1 q. (2.15). m 1. Thanks to Lemma 2.1.1 we estimate ¸. ¥. f pmqNΓ px, mq. ¤. m z. which becomes opLipxqq taking z ¸. . f pmqNΓ px, mq. ¤. y m z. maxt|f pmq|u OΓ. ¤. m x. x. . r 2 δr r 1. ¸. . . px{zq1 1{r ! x1δ px{zq1 1{r , logpx{z q logpx{z q. . Finally, the last estimate we need is. |f pmq| πpx; 1, mq !. y m z. | p q| Lipxq opLipxqq , ϕpmq m ¡y ¸ f m. where we have used the Brun-Titchmarsh Theorem together with the convergence of the series ° | m¥1 f pmq|{ϕpmq. The following Theorem gives an unconditional estimate for the sum. °. ¤ f pip q.. p x p-σΓ. Theorem 2.2.5. Let Γ Q be a finitely generated subgroup of rank°r ¥ 2 and let f : N Ñ C be ° an arithmetic function with frpnq d|n µpn{dqf pdq. Suppose that n¥1 |frpnq|{ϕpnq converges and that. °. 1 ¡ |frpnq|{n oplog xq, then. n x. ¸. ¥. frpnqπΓ px, nq . n 1. p q Lipxq ,. ¸ fr n. k pΓq n¥1 n. as x Ñ 8. Proof. We first perform the following splitting, ¸. ¥. n 1. frpnqπΓ px, nq . ¸. ¤. frpnqπΓ px, nq. ¸. ¡. n y. n y. 13. frpnqπΓ px, nq ,. (2.16). . q. 3. ,.

(27) 1{p3r. . q. x and c1 is the constant appearing in where, as in the previous proof, y c1 ploglog log xq2 unconditional statement of Theorem 1.2.3. Hence, the first sum is. ¸. ¤. ¸. frpnqπΓ px, nq . n y. ¤. frpnq. . n y. Lipxq kn pΓq. . OΓ . k ppΓqq Lipxq n¥1 n ¸ fr n . OΓ. . 3. . x exp c2 plog xq { plog log xq1{3 1 6. | p q| Lipxq k pΓq n¡y n. O. ¸ fr n. . x log x 1{6 1{3 exp c p log x q p log log x q 2 plog log xq2. p q Lipxq. ¸ fr n. . opLipxqq ,. k pΓq n¥1 n. where the assumption (2.6) has been used. To conclude the proof, we split the second sum as ¸. ¡. frpnqπΓ px, nq . n y. ¸. ? y n¤ x. frpnqπΓ px, nq. ¸. frpnqπΓ px, nq ;. ? n¡ x. since πΓ px, nq ¤ π px; 1, nq #tp ¤ x : n | p 1u, by the Brun-Titchmarsh Theorem we have ¸. ¤?x. frpnqπΓ px, nq !. y n. ¸. ¤?x. y n. x |frpnq| ϕpnq log px{nq opLipxqq ,. while ¸. ¡?x. n. 2.3. frpnqπΓ px, nq ¤. ¸. ¡?x. |gpnq|#tm ¤ x : n | m 1u !. n. ¸. ¡?x. n. |frpnq| nx opLipxqq .. Examples and applications. In this section we want to generalize some applications in [23] to the higher rank case: we will do it in the case Γ Q , where Q denotes the set of positive rational numbers. We first compute the densities corresponding to the cases f pnq multiplicative and totally multiplicative function respectively, where f : N Ñ C is the arithmetic function appearing in the sum (2.1). Lastly we will apply those results to the case of f pnq χB pnq, the characteristic function of certain subsets B N, and f pnq χpnq mod b, a Dirichlet character modulo b. 14.

(28) 2.3.1. f pnq multiplicative function. Suppose that the arithmetic function f pnq in (2.1) is multiplicative; consequently, also frpnq is multiplicative. We want to compute the density δΓ,f :. pq. k pΓq n¥1 n ¸ fr n. (2.17). From now on, we will restrict to the case Γ Q , so that we can make use of Lemma 1.2.2. In order to compute the density (2.17) we begin splitting it as a sum over odd integers, So , plus a sum over even integers, Se : δΓ. So. Se. frpnq |Γrpnq| ϕ p n q| Γ p n q| n¥1. frpnq |Γrpnq| . ϕ p n q| Γ p n q| n¥1. ¸. . ¸. |. 2-n. 2n. To lighten the next formulas, we denote hpnq :. frpnq , ϕpnq|Γpnq|. r pn q which is a multiplicative function since frpnq, ϕpnq and |Γpnq| are. Noticing that Γ whenever n is odd, the sum over odd integers has the following Euler product:. So. . ¸. ¥. hpnq . n 1 2-n. ¹. . ¥. ¸. ¥. p 3. hppk q. . 1. .. k 0. For every m | σΓ we write ∆pmq 2v2 p∆pmqq m1 ; hence the sum over even integers can be written as Se. . ¸. ¥. n 1 2-n. hpnq. ¸. P pq. rn m Γ. 1. ¸. |. m σΓ. . ¸. ¥α. ¸. ¥ α1 2 m2 Q PΓp2α q pp q q| α 1. ¸. ¸. |. α 1. m σΓ m2. ¥α. n 1 v2 n α ∆ m n. hp2. α. q. ¥ p p qq. ¹. . ¸. |. α 1. m σΓ m2. . ¥α. hp2α q. ¹. ¥ |. p 3 p m1. Q Γ 2α α v2 ∆ m. 2 P p q ¥ p p qq. α 1. . ¥. α v2 ∆ m. ¸. 15. ¸. ¸. |. α 1. m σΓ m2. hp2 α q. ¥α. Q2 PΓp2α q. α 1. ¸. hpp. k. q. ¥ p p qq. 1. ¹. ¥ |. p 3 p m1. k 0. . . ¸. ¥. j 1. hppj q. . ¸. ¥. hpp. j. j 1. 1 1 . ¸. ¥ |. n 1 2-n m1 n. α v2 ∆ m. ¥. p 3 p-m1. Q2 PΓp2α q. α 1. hpn q . .. q. hpn q.

(29) We define tm :. #. mintz. 8. P N Y t 0u :. and γm : maxt1 $ ' ' &. δΓ,f. So '1 ' %. . ¹. . ¸. P Γp2z 1qu. z 1. z. if Dz P N such that m2 otherwhise ,. z. P Γp2z 1q ,. tm , v2 p∆pmqqu, so that ¸. ¸. |. ¥. hp2α q. m σΓ α γm. hppk q. ¥. p. m2 Q 2. ¹. ¥ |. , 1 1 / / . ¸ j hp / / j ¥1 -. . . 1. p 3 p m1. $

(30) 1 ' ' & ¸ β h2 1 ' ' β ¥0 %. . k 0. p q. p q. ¸. ¸. |. ¥. hp2α q. m σΓ α γm. ¹. ¥ |. , 1 1 / / . ¸ j hp . / / j ¥1 -. . . 1. p 3 pm. p q. Rearranging the terms in the previous expression we arrive at. δΓ,f. ¹. . . ¸. hppk q. ¥. p. k 0. $ ' ' &. °. ¸. 1. ' ' %. . ¥ hp2 q ¹ 1 β β ¥1 hp2 q p|2m α. α γm. °. | . m σΓ m 1. . , 1 1 / / . ¸ j hp . / / j ¥1 -. p q. In order to express δ in terms of the function f pnq, we note that frppk q f ppk q f ppk1 q; ° Γ,f moreover, the sum α¥γm hp2α q becomes zero if γm 8. Hence we obtain δΓ,f. . ¹. . 1. p. $ ' ' ' ' &. ¹. |. q 2m. | 8. 1. | p. |Γp12q|. m σΓ m 1 γm. . p. p q. ¸ f pk. p. p 1 k¥1 pk. f 2γm 1 γm 1 Γ 2γm. 2. ¸. 1. ' ' ' ' %. 1 pp 1q|Γppq|. q. q|. 2 2. °. °. ¥. β 1. . pq 11q|Γpqq|. q. p q. f 2α 2α. ¥ f p2β q. α γm. 1 |Γppk q|. 2β. 1 Γ 2β. | p. p q. ¸ f qj. q 1 j ¥1. . qj. 1 Γ 2α. | p. . 1 p|Γppk. 1 q| 2|Γp2. 1. α 1. 1 q| 2|Γp2. β 1. . . 1 |Γpqj q|. q|. q|. . q|. . q|Γpq1j 1q|. ,. 1 1 . . -. This formula can be made more compact introducing appropriate quantities, as we do in the following: 16.

(31) Theorem 2.3.1. Given a subset Γ then. δΓ,f. . ¹. p1. Ap q. p. Q. of finite rank r and a multiplicative function f pnq,. $ ' ' ' ' &. . ¸. 1. ' ' ' ' %. 1. | 8. m σΓ m 1 γm. where Ap. . AΓ,f p. pp 11q|Γppq|. ¹. |. p 1 k¥1 pk. BmΓ,f 2. 1 |Γppk q|. f p2α1 q f p2α q. γm ¸1. 2α |Γp2α q|. . 1 A q. 1. 1. q 2m. p q. α 1. Let us focus on the case r following:. ¸ f pk. p. and Bm. Bm A2. . , / / / / . / / / / -. 1 p|Γppk. ,. ,. (2.18). 1. q|. (2.19). (2.20). 1, i.e. suppose Γ xay with a P Qzt1u. We first state the. Lemma 2.3.2. Let Γ xay with a P Q zt1u and let h be the largest integer such that a bh for some rational number b b1 b22 , with b1 P Z unique squarefree integer with such a property. For any m 1 squarefree positive integer we have tb1 v2 phq and tm 8 if m b1 . Proof. First note that in this case h n b Q Q n . |xaypnq| x y Since |xayp2z 1 q| 2z 1 {p2z 1 , hq 2maxt0,z m 1. Also note that, if we write h h1 2v2 phq then a b12. pn,nhq .. p qu , we have that tm ¥ v2 phq for all integers. 1 v2 h. 2v phq. h2 2v2 phq. 2. p q h2 h1 2v2 phq b1 b2. a pQ q2. pq. v2 h. We deduce that. 1. b21. xayp2v phq 1q tpQq2 2. pq. v2 h. pq. v2 h. 17. 1. pQq2. , b21. pq. v2 h. pq. v2 h. ,. 1. v2 h. so that. 1. 1. pQq2. .. pq. v2 h. 1. u..

(32) which implies that tb1. v2phq. More in general, for z v2phq !. xayp2z 1q . aj. pQq2. z 1. : j. P Z{2k. k and k 1. ¥ 0,. ). .. Z. For m 1 square free, the identity z. m2. pQq2. z 1. is satisfied if and only if j. aj pQq2. z 1. b1j2. p q h2 h1 j2v2 phq b1 b2. v2 h. 1. pQq2. z 1. 2k and m b1.. Lemma 2.3.2 leads to the following: Theorem 2.3.3. Let Γ xay with a P Q zt1u, where h is the largest integer such that a and b b1 b22 , with b1 P Z unique squarefree integer with such a property. Then δxay,f. . ¹. p1. Ap q. p. $ & %. . 1. Bb1 A2. 1. ¹. |. 1. pA1q q. , . 1 1. q 2b1. -. ,. bh. (2.21). with Ap Axay,f p. ppppp,hq1q. p q ppk , hq ppk 1, hq. p2 p2k. ¸ f pk. p. p 1 k ¥1. (2.22). and. xay,f Bb1 Bb1. 2. ¸ . α 1. with γb1. rf p2α1q f p2αqs p222α, hq ,. γb1 1. α. (2.23). 1 maxtv2phq, v2p∆paqq 1u.. We can now easily retrieve the classical result of Hooley [9, Theorem] from Theorem 2.3.3 in the case of positive rationals, choosing f pnq χt1u pnq: Theorem 2.3.4 (Hooley). Consider a positive rational a 0 which is not a perfect square and let h be the largest integer such that a bh ; denote with d the discriminant of the quadratic ? extension Qp aq. Assuming GRH for the number field Qpζn , a1{n q for every squarefree n, then Na pxq δa Lipxq 18.

(33) as x Ñ 8, where µ pn q δa rQpζn, a1{nq : Qs n¥1 ¸. $ Ah ' &. pq. ± ± 1 µp|d|q q|d q1 2 q|d q 1q1 Aphq. ' %. ¹. Aphq . . 1. q-h. 1. ppp 1q. p-h. if d 1 p mod4q ,. 2. |. qh. with. if d 2, 3 p mod4q ,. ¹. |. . 1. ph. 1. p1. .. In general, the expressions (2.19) and (2.20) cannot be further simplified, unless we make additional suppositions on Γ and consequently on |Γppk q|. An interesting case is the following: Corollary 2.3.5. Suppose Γ xp1 , . . . , pr y, with pi generic prime number, then for any multiplicative function f pnq we have. δxp1 ,...,pr y,f. where Ap. . ¹. Ap q. p. ' ' %. . ¸. 1. Bm A2. 1. | . m σΓ m 1. ¹. |. 1. 1 A q. 1. q 2m. , / / . / / -. r 1 k ¸ Apxp ,...,p y,f pp 11qpr ppp 1qp1r pfkpprp 1qq k ¥1. Bm B xp1 ,...,pr y,f. r. 1. and m. p1. $ ' ' &. . $ ' ' &0. pq. 1 f 2 2r ' ' 1 % r 1 2. f p2q. . . 2r 1 1 2r 1. . pq. f 4 2r 1. °. . if m 2 p mod4q .. and Bm B xp1 ,...,pr y,f m. . $ ' ' &0. q. 1 k. (2.24). (2.25). if m 1 p mod4q , if m 3 p mod4q ,. Moreover, if f pnq is totally multiplicative and the series k¥1 pf ppq{pr p, then ppf ppq 1q Ap Axpp1 ,...,pr y,f pp 1qppr 1 f ppqq. ,. (2.26). converges for every (2.27). if m 1 p mod4q ,. pq if m 3 p mod4q , . p q 1 f p2q if m 2 p mod4q . 2r 1. 1 f 2 2r ' ' 1 % fr 2 2. 19. (2.28).

(34) Proof. It is sufficient to apply Theorem 2.3.1 with |Γppk q| Moreover, since tm 0 for every m | σΓ , notice also that $ ' &1. γm. '3 %. 2. pkr because of [2, Proposition 2].. if m 1 p mod4q , if m 2 p mod4q , if m 3 p mod4q .. Notice that the previous Corollary holds, in general, for all the subgroups Γ Q such that |Γpnq| nr for every natural number n.. 2.3.2. f pnq χB pnq characteristic function of a subset B. Let χB be the characteristic function of a subset B πΓ,B pxq : #tp ¤ x : p - σΓ , ip. N. N, then defining ¸. P Bu . ¥. χB pmq NΓ px, mq. m 1. by Möbius inversion formula we obtain πΓ,B pxq . ¸. ¥. χ rB pmq πΓ px, mq ,. m 1. where. rB pmq χ. ¸. |. µpm{dq χB pdq . dm. Define δΓ,B :. ¸. | P. µpm{dq .. dm d B. p q; k pΓq n¥1 n ¸ χ rB n. then, assuming GRH, we have from Theorem 2.2.1 that πΓ,B pxq δΓ,B Lipxq as x Ñ 8. We will also see explicitly some examples where the unconditional results from Theorems 2.2.4 and 2.2.5 can be applied. 20.

(35) χP pnq characteristic function of the primes. Let B P be the set of prime numbers. Applying Theorem 2.2.1 we obtain the asymptotic density result appeared in [24, Theorem 1.1] in the case Γ xa1 , . . . , ar y, with a1 , . . . , ar multiplicatively independent integers different from 0, 1 and not all perfect squares. Lemma 2.3.6. Under GRH, if Γ Q is a finitely generated subgroup of rank r, then NΓ pxq #tp ¤ x : p - σΓ , ip where δΓ,P. . 1u δΓ,P Lipxq , p q. ¸ µ m. k pΓq m ¥1 m. as x Ñ 8 ,. .. χB pnq multiplicative function. Suppose B N is such that χB pnq is multiplicative, then we can apply Theorem 2.3.1 with f pnq χB pnq to get the following: Theorem 2.3.7. Consider the r-rank subgroup Γ Q and a subset B characteristic function χB pnq is multiplicative; assuming GRH we have πΓ,B pxq δΓ,B Lipxq ,. . N such that its. xÑ8,. where. δΓ,B. . ¹. p1. Ap q. p. $ ' ' ' ' & ' ' ' ' %. . ¸. 1. | 8. 1. m σΓ m 1 γm. Bm A2. ¹. |. 1 A q. 1. q 2m. 1. , / / / / . / / / / -. ,. (2.29). where Ap. AΓ,B pp 11q|Γppq| p. p 1 k¥1. and Bm. . Γ,B Bm. 2. p q. ¸ χB pk. p. pk. 1 |Γppk q|. p|Γpp1k 1q|. χ p2α1 q χ p2α q B B. γm ¸1. . α 1. 2α |Γp2α q|. .. (2.30). (2.31). We can apply the previous results in the simpler case Γ xp1 , . . . , pr y to get the following: 21.

(36) Corollary 2.3.8. Let Γ xp1 , . . . , pr y, with pi generic prime number. For every subset B such that its characteristic function χB pnq is multiplicative, assuming GRH we have πΓ,B pxq δxp1 ,...,pr y,B Lipxq ,. N. xÑ8,. with. δxp1 ,...,pr y,B. where Ap. . ¹. p1. $ ' ' &. Ap q. ' ' %. p. . ¸. 1. 1. | . m σΓ m 1. Bm A2. ¹. |. q 2m. r 1 Axpp ,...,p y,B pp 11qpr ppp 1qp1r. and Bm B xp1 ,...,pr y,B m. ¸. r. 1. ¥ P. pq. χB p2 q. . . 2r 1 1 2r 1. χ2 p4q B r 1. 1. k 1 pk B. $ ' ' &0. 1 χB 2 2r ' ' 1 % r 1 2. 1. 1 A q. 1. . p. , / / . / / -. ,. (2.32). q. pk r 1. if m 1 p mod4q , if m 3 p mod4q , if m 2 p mod4q .. Moreover, if χB pnq is totally multiplicative, then #. Ap. 0 1 p pp1q r. and Bm. . #. if p P B , if p R B ,. if m 1 p mod4q or 2 P B , if m 2, 3 p mod4q and 2 R B .. 0 1 2r. Let us consider some examples in a more detailed way. Example 1. Let Hk tn P N : n mk , m P Nu be the set of k-powers. For k ¸. P. n Hk. 1 ϕpnq. . ¥ 2, the sum. ¸. 1 ϕpmk q m¥1. converges, so that from Theorems 2.2.4 and 2.2.2 we derive the following: Corollary 2.3.9. Let Γ Q be a finitely generated subgroup and let Hk tn P N : n mk , m P Nu be the set of k-powers, with k ¥ 2. Then, as x Ñ 8, πΓ,Hk pxq À δΓ,Hk Lipxq unconditionally and πΓ,Hk pxq Á δΓ,Hk Lipxq under GRH. 22.

(37) In the case Γ xp1 , . . . , pr y, formula (2.32) can be applied with Ap. kpr 1qr Axpp ,...,p y,H pp p 1qppkpr 1q p 1q r. 1. and Bm B xp1 ,...,pr y,Hk m. . k. $ ' 0 ' &. 1 2r ' ' % 1r 2. . 1. if m 1 p mod4q , if m 3 p mod4q ,. p q. if m 2 p mod4q .. χHk 4 2r 1. In Table 2.1 we compute the density values δΓ,Hk for the cases k 2, 3, 4 and Γ xpi , pj y, with p 237 , computed distinct pi , pj P t2, 3, 5, 7, 11, 13u, and compare them with the numerical results δΓ,H k 37 1 up to p 2 , using the software GP/PARI CALCULATOR Version 2.7.5 . Example 2. Let k ¥ 2, let Fk tn P N : vp pnq k, @p primeu be the set of k-free numbers. Since # µpmq if n mk , rFk pnq χ 0 otherwise , the conditions of Theorem 2.2.5 are satisfied, so that we can state the following: Corollary 2.3.10. Let Γ Q be a finitely generated subgroup and let Fk tn P N : vp pnq k, @p primeu be the set of k-free numbers, with k ¥ 2. Then, as x Ñ 8, πΓ,Fk pxq δΓ,Hk Lipxq unconditionally. If Γ xp1 , . . . , pr y, formula (2.32) holds with Ap. Axpp ,...,p y,F pkpr 1. r. and Bm B xp1 ,...,pr y,Fk m. 2.3.3. . 1. k. #. 0. . pq. 1 χFk 4 22r 1. 1. q1 pp 1q. if m 1, 3 p mod4q , if m 2 p mod4q .. Indices ip in arithmetic progression. Given two coprime positive integers a and b, consider the set SΓ px; a, bq : tp ¤ x : p - σΓ , ip 1. a pmod bqu. Together with Henri Cohen’s script (see http://pari.math.u-bordeaux.fr/Scripts/), which includes the function prodeuleratt for calculating Euler products.. 23.

(38) Table 2.1: Comparison between the theoretical (GRH) δΓ,Hk and numerical (up to p p 237 , where Γ xpi , pj y, pi , pj P t2, 3, 5, 7, 11, 13u. density δΓ,H k δΓ,H2 δΓ,H3 δΓ,H4 Γ p 237 p 237 p 237 δΓ,H2 δΓ,H3 δΓ,H 4 0.7204017835 0.7035521973 0.6982514901 x2, 3y 0.7204008108 0.7035514839 0.6982504751 0.7062105809 0.7006387110 x2, 5y 0.7237918508 0.7237946933 0.7062128444 0.7006414714 0.7039046595 0.6982955321 x2, 7y 0.7216069808 0.7216104694 0.7039091705 0.6983004502 0.7039211320 0.6982975876 x2, 11y 0.7216653483 0.7216666678 0.7039209604 0.6982978180 0.7040379691 0.6984125550 x2, 13y 0.7217878018 0.7217875330 0.7040386343 0.6984125509 0.7028723530 0.7002221463 x3, 5y 0.7255448866 0.7255450297 0.7028728547 0.7002227099 0.7004972642 0.6978294779 x3, 7y 0.7233268507 0.7233301700 0.7004996641 0.6978328171 0.7005403348 0.6978658086 x3, 11y 0.7234237628 0.7234241660 0.7005412999 0.6978673966 0.7006673436 0.6979919473 x3, 13y 0.7235577077 0.7235598185 0.7006681588 0.6979936959 0.7030468041 0.7002439449 x5, 7y 0.7269084536 0.7269133076 0.7030492041 0.7002473531 0.7030549572 0.7002449623 x5, 11y 0.7269744910 0.7269774849 0.7030499313 0.7002449623 0.7031695617 0.7003586371 x5, 13y 0.7270967856 0.7270997051 0.7031721896 0.7003617319 0.7007270610 0.6978984668 x7, 11y 0.7248102383 0.7248161001 0.7007316472 0.6979033861 0.7008434877 0.6980139575 x7, 13y 0.7249343967 0.7249362846 0.7008475860 0.6980175153 0.7008517199 0.6980149847 x11, 13y 0.7250010696 0.7250008316 0.7008527808 0.6980159664. . 237 ). with cardinality NΓ px; a, bq #SΓ px; a, bq. Let χ : N Ñ C be a Dirichlet character modulo b, then the ortogonality relations can be used to define the following characteristic function: χa,b pnq :. ¸ 1 χpa qχpn q ϕpbq χ mod b 24. #. 1 0. if n a (mod b) otherwhise ..

(39) Consider the density δΓ,pa,bq : δΓ,SΓ pa,bq. p q 1 ¸ χpa qδ , χ kn pΓq ϕpbq χ mod b. ¸ χ ra,b n. . ¥. n 1. where δχ. . pq. k pΓq n¥1 n ¸ χ r n. (2.33). From Theorem 2.2.1 follows the following: Corollary 2.3.11. Let Γ Q be a finitely generated subgroup. On GRH we have, as x Ñ 8, NΓ px; a, bq δΓ,pa,bq Lipxq . A Dirichlet character χpnq is a totally multiplicative function, so in the case Γ Q we can compute the density (2.33) making use of formula (2.18). In the specific case Γ xp1 , . . . , pr y we can apply formulas (2.24), (2.27) and (2.28) to get δxp1 ,...,pr y,pa,bq. ϕp1bq $ ' ' &. ¸. ¸. 1. Ap. | . Ap q . . 1. Bm A2. ¹. |. 1. 1 A1 q. q 2m. , / / . / / -. (2.34) ,. Axpp ,...,p y,χ pp p1pqpχpppr q 11qχppqq 1. Bm B xp1 ,...,pr y,χ . 2.3.4. p1. p. m p1 pr m 1. and m. ¹. χ mod b. ' ' %. with. χpaq. r. $ ' ' &0. (2.35). if m 1 p mod4q ,. pq if m 3 p mod4q , . p q 1 χp2q if m 2 p mod4q . 2r 1. 1 χ 2 2r ' ' 1 % χr 2 2. (2.36). Average order of Γp over primes. In the special case when Γ xg y, where g P Q zt1u, Kurlberg and Pomerance [11, Theorem 2] proved under GRH an asymptotic formula for the average over primes up to x of the order op pg q |xg pmod pqy|: cg 1 ¸ o p pg q x Lipxq p¤x 2. . O 25. x. plog xq24{ log log log x. ,.

(40) as x Ñ 8, where. cg :. pq. p qp1qωpnq . n2 kn pxg yq. ¸ ϕ n Rad n. ¥. n 1. For a finitely generated subgroup Γ Q of rank r, the previous result has been generalized by Pehlivan in [27]:. ¸ log log x 1 t (2.37) |Γp| cΓ,t OΓ,t plog xqr , Lipxt 1 q p¤x. as x Ñ 8, where. p qqtp1qωpnq n2t kn pxg yq n¥1 ± with the Jordan’s totient function defined by Jt pnq : nt q|n p1 1{q t q. We can retrieve the main term in (2.37), as x Ñ 8, choosing the function f pnq 1{nt and, starting with the sum cΓ,t :. p qp. ¸ Jt n Rad n. 1 Lipxt. ¸ 1. 1. q p¤x itp ,. all we need to do is to perform a summation by parts after writing 1 itp. . p q.. ¸ µ a. |. ab ip. 26. bt.

(41) Chapter 3. Average r-rank Artin’s conjecture The results discussed in this Chapter appears in the joint work of the author with Cihan Pehlivan [17].. 3.1. Introduction. In the case of rank r 1, a first unconditional result on the average behavior of Napxq was pre ± sented by Goldfeld [6] in 1967, who proved that for any constant D ¡ 1, if A p 1 ppp11q indicates the Artin’s constant, the asymptotic formula . Na pxq A Lipxq. O. holds for all integers a ¤ N with at most c1 N 9{10 p5 log x. 1qg. D 2. x plog xqD. ,. g. log x log N. ,. exceptions, where c1 and the constant implied by the O-notation are positive and depend at most on D. Stephens [31] in 1969 proved that, if T ¡ expp4plog x log log xq1{2 q, then ¸ ϕpp 1q 1 ¸ Na pxq T a¤T p1 p ¤x. . O. x plog xqD. . A Lipxq. O. x plog xqD. ,. (3.1). where ϕ is the Euler totient function and D is an arbitrary constant greater than 1. Stephens also proved that, if T ¡ expp6plog x log log xq1{2 q, then 1 ¸ tNapxq A Lipxqu2 T a¤T 27. 2. ! plogxxqD1. ,. (3.2).

(42) for any constant D1 ¡ 2. In 1976, Stephens refined his results with different methods [32], getting both the asymptotic bounds (3.1) and (3.2) under the weaker assumption T ¡ exppC plog xq1{2 q, with C positive constant. If we set, for any a P Nzt0, 1u and m P N, Na,m pxq to be the number of primes p 1 pmod mq not exceeding x such that the index rFp : xa pmod pqys m, then for T ¡ expp4plog x log log xq1{2 q Moree [21] showed that 1 ¸ Na,m pxq T a¤T. ¸. ¤. p x p 1 mod m. p. q. ϕppp 1q{mq p1. for any constant E ¡ 1. Here we restrict ourselves to studying subgroups Γ 1, . . . , r, and we prove the following Theorems: Theorem 3.1.1. Let T : mintTi : i 1, . . . , ru for an arbitrary positive constant D. Then ¸. 1. T1 Tr. P ¤. . O. x plog xqE. ,. (3.3). xa1, , ar y, with ai P Z for all i . ¡ expp4plog x log log xq q and m ¤ plog xqD. Nxa1 , ,ar y,m pxq Cr,m Lipxq. 1 2. . O. ai Z 0 a1 T1. x plog xqM. ,. .. .. ¤. 0 ar Tr. where Cr,m : and M. µ pn q pnmqr ϕpnmq n¥1 ¸. ¡ 1 is arbitrarily large.. Theorem 3.1.2. Let T constant D. Then 1. T1 Tr. ¡ expp6plog x log log xq q and m ¤ plog xqD for an arbitrary positive 1 2. ¸. P ¤. Nxa1 , ,ar y,m pxq Cr,m Lipxq. ai Z 0 a1 T1. .. .. ¤. 0 ar Tr. where M 1. ¡ 2 is arbitrarily large. 28. (2. 2. ! plogxxqM 1.

(43) Now since ϕpmnq ϕpmqϕpnq gcdpm, nq{ϕpgcdpm, nqq and gcdpm, nq is a multiplicative function of n for any fixed integer m, we also have the following Euler product expansion: Cr,m. . ¸ µp n q 1 mr ϕpmq n¥1 nr ϕpnq. | p. p gcd m,n. . ¹ 1 1 1 r r m ϕpmq p pp 1 q. |. pm. mr. ¹. 1 1. |. . 1. pm. pr pr. 1. . ¹. 1. 1. q. . 1. 1 p. 1 p. . ¹. . 1. p-m. 1. 1 r p p p 1q. Cr .. The results found in the present paper (see in particular equation (3.8) and Lemma 3.3.3) will lead as a side product to the asymptotic identity 1. T1 Tr. ¸. P ¤. ¸. Nxa1 , ,ar y,m pxq . ¤. p x p 1 mod m. ai Z 0 a1 T1. p. .. .. Jr ppp 1q{mq pp 1 qr. q. . O. x plog xqM. ,. ¤. 0 ar Tr. if Ti ¡ expp4plog x log log xq 2 q for all i where 1. 1, . . . , r, m ¤ plog xqD and M ¡ 1 arbitrary constant,. Jr pnq n. r. ¹. |. `n ` prime. . 1. 1 `r. is the so called Jordan’s totient function. This provides a natural generalization of Moree’s result in [21]. Theorem 3.1.2 leads to the following Corollary: Corollary 3.1.3. For any ¡ 0, let H : ta P Zr : 0 ai. ¤ Ti, i P t1, . . . , ru, |Na,mpxq Cr,m Lipxq| ¡ Lipxqu ; then, supposing T ¡ expp6plog x log log xq1{2 q, we have #H ¤ K |T |{2 plog xqF for every positive constant F . Proof of Corollary 3.1.3. The proof of this Corollary is a trivial generalization of that in [31] (Corollary, page 187). 29.

(44) 3.2. Notations and conventions. In order to simplify the formulas, we introduce the following notations. Underlined letters stand for general r-tuples defined within some set, e.g. a pa1 , . . . , ar q P pFp qr or T pT1 , . . . , Tr q P pR¡0qr ; moreover, given two r-tuples, a and n, their scalar product is a n a1n1 ar nr . The null vector is 0 t0, . . . , 0u. Similarly, χ pχ1 , . . . , χr q is a r-tuple of Dirichlet characters and, given a P Zr , we denote the product χpaq χ1 pa1 q χr par q P C. In addition, pq, aq : pq, a1 , . . . , ar q gcdpq, a1 , . . . , ar q; otherwise, to avoid possible misinterpretations, we will write explicitly gcdpn1 , . . . , nr q instead of pnq. Given any r-tuple a P Zr , we indicate with xayp : xa1 pmod pq, . . . , ar pmod pqy. the reduction modulo p of the subgroup xay xa1 , . . . , ar y Q; if Γ xa1 , . . . , ar y, then Γp xayp . In the whole paper, ` and p will always indicate prime numbers. Given a finite field Fp , then x will denote its relative dual group (or character group). Finally, given an Fp Fp zt0u and F p integer a, vp paq is its p-adic valuation.. 3.3 Let q. Lemmata. ¡ 1 be an integer and let n P Zr . We define the multiple Ramanujan sum as ¸ e2πian{q . cq pnq : Pp { q p q. a Z qZ r q,a 1. It is well known (see [8, Theorem 272]) that, given any integer n, cq pnq µ. . q pq, nq. ϕ pq q. . q q,n. .. p q. ϕ. In the following Lemma, we generalize this result to r-rank. Lemma 3.3.1. Let. Jr pmq : m. r. ¹. . |. 1. `m. 1 `r. be the Jordan’s totient function, then cq pnq µ. . q pq, nq 30. Jr p q q . Jr. q q,n. p q. .. (3.4).

(45) Proof. Let us start by considering the case when q c` pnq. . ¸. Pp { qr zt0u. e2πian{`. a Z `Z. ` r ¸ ¹. 1. ` is prime. Then. . . #. r1 ` 1. e2πiaj nj {`. j 1 aj 1. Next we consider the case when q. if ` - gcdpn1 , , nr q , otherwise.. `k with k ¥ 2 and ` prime. We need to show that. $ ' &0. if `k1 - gcdpn1 , , nr q , c`k pnq `rpk1q if `k1 } gcdpn1 , , nr q , ' % rk ` 1 `1r if `k | gcdpn1 , , nr q . To prove that, we start writing c`k pnq. . ¸. Pp { qr p q. e2πian{`. k. a Z `k Z `,a 1. . c`k pn1 q. k. r ¸ ` ¹. . . e2πiaj nj {`. j 2 aj 1. . c`k pn1 q. k. ` r ¸ ¹. . . k ¸. c`k pn2 , . . . , nr q. k. . ¸. e2πia1 n1 {`. k. P { q. j 1 a1 Z `k Z a1 ,`k `j. e2πiaj nj {`. k ¸. c`k pn2 , . . . , nr q. k. j 2 aj 1. p. . c`kj pn1 q .. j 1. If we apply (3.4), we obtain c`k pn1 , . . . , nr q. . . µ. `k p`k , n1q. ϕp`k q. . ϕ. c`k pn2 , . . . , nr q. p. `k `k ,n1. k ¸. . j 1. ¥ 2, let us distinguish the two cases: `k1 - gcdpn1 , . . . , nr q , `k1 | gcdpn1 , . . . , nr q .. Now, for k 1. 2.. 31. µ. k. q. . ` r ¸ ¹. . . e2πiaj nj {`. k. j 2 aj 1. `kj p`kj , n1q. ϕp`kj q . k j ϕ p`k`j ,n q 1 .

(46) . k In the first case we can assume, without loss of generality, that - n1 . Hence µ p`k`,n q 1. and if k1. `k 1. v`pn1q k 1, then . µ. Hence. . `k j. t uq . p`kj , n1q µp`. k ¸. . . µ. j 1. max 0,k k1 j. `kj p`kj , n1q. ϕp`kj q. k j ϕ p`k`j ,n q 1 . $ ' &0 ' %. 0. if 1 ¤ j ¤ k k1 2, if j k k1 1, if j ¥ k k1 .. 1. 1. k ¸. `k. 1. . ϕp`kj q 0.. j k k1. In the second case, from the definition of cq pnq we find c`k pnq ` p q c` r k 1. n 1. nr . , . . . , k 1 `k1 `. . #. `rk 1 `1r `rpk1q. . if `k | gcdpn1 , . . . , nr q , if `k1 } gcdpn1 , . . . , nr q .. So, the formula holds for the case q `k . Finally, we claim that if q 1 , q 2 P N are such that gcdpq 1 , q 2 q 1, then cq1 q2 pnq cq1 pnq cq2 pnq ; this amounts to saying that the multiple Ramanujan sum is multiplicative in q. Indeed °. Pp { q e2πian{q p q. a Z q1 Z r q 1 ,a 1. ¸. 1. b Z q2 Z r q 2 ,b 1. Pp { q p q. . e2πibn{q. ¸. a Z q1 Z r b Z q2 Z r gcd q 1 ,a 1 gcd q 2 ,b 1. Pp { Pp { p p. q q q q. 2. e2πirn1 pq. 2 a1 q1 b1 q . nr q 2 ar q 1 br. p. qs{pq1 q2 q. and the result follows from the remark that, since gcdpq 1 , q 2 q 1,. • for all j 1, . . . r, as aj runs through a complete set of residues modulo q 1 and as bj runs through a complete set of residues modulo q 2 , q 2 aj q 1 bj runs through a complete set of residues modulo q 1 q 2 . • for all a P pZ{q 1 Zqr and for all b P pZ{q 2 Zqr , gcdpq 1 , aq 1. and gcdpq 2 , bq 1. ðñ. gcdpq 1 q 2 , q 1 b1 32. q 2 a1 , . . . , q 1 br. q 2 ar q 1..

(47) The proof of the Lemma now follows from the multiplicativity of µ and of Jr . From the previous Lemma we deduce the following Corollary: Corollary 3.3.2. Let p be an odd prime, let m P N be a divisor of p χ pχ1 , . . . , χr q of Dirichlet characters modulo p, we set cm pχq :. ¸. 1. pp 1 qr. α F p F p: α p. Pp qr r x y sm. 1.. Given a r-tuple. χpα q .. Then . cm pχq . 1. pp 1 q. r. µ. . m gcd. p1. . , . . . , p 1 p q ordpχr q. p 1 p 1 m , ord χ1. Proof. Let us fix a primitive root g. ga. j. for j. Therefore, naming t . . . Jr. m gcd. . Jr. p 1 m. . p 1. . ,..., p1 p q ordpχr q. .. (3.5). p 1 , p 1 m ord χ1. P Fp . For each j 1, . . . , r, let nj P Z{pp 1qZ be such that χj. if we write αj. .

(48). χj pg q e. 2πinj p 1. . ;. 1, . . . , r, then rFp : xαyps m ðñ pp 1, aq m .. we have. p 1 m ,. cm pχq . ¸. 1. pp 1qr. pp 1 1qr. r aPpF pq pp1,aqm. ¸. a1. PpZ{tZqr pt,a1 q1. χ1 pg qa1 χr pg qar. 1 e2πia n{t. pp 1 1qr c pnq. p 1 m. By definition we have that ordpχj q pp 1q{ gcdpnj , p 1q, so p1 . m gcd. . p 1 m ,n. . . m gcd. . p1. , . . . , p 1 p q ordpχr q. p 1 p 1 m , ord χ1. and this, together with Lemma 1, concludes the proof. 33. (3.6).

(49) For a fixed rank r, define Rp pmq : #ta P pZ{pp 1qZqr : pa, p 1q well-known properties of the M¨ obius function, we can write Rp pmq . ¸ . ¸. P ppZ1qZ. r. a. |p ma , pm1 q. hm pnq # a P. so that. Rp pmq . . n. n. ". where. p1 m. ¸. µp n q . | pm1. a Z :n| p p 1 qZ m. r. *. mu. Then using. µpnqrhm pnqsr ,. 1 , pnm. p q J p 1 . r nr m. ¸ µ n. | pm1. n. (3.7). Defining Sm p xq :. 1 mr. ¸. ¤. p1 q |m. p x n p 1 mod m. p. p q nr. ¸ µ n. ¸. ¤. p x p 1 mod m. p. . 1. q. pp 1 qr J r. p1 m. ,. (3.8). we have the following Lemma: Lemma 3.3.3. If m constant M ¡ 1. where Cr,m. ¤ plog xqD , with D arbitrary positive constant, then for every arbitrary . Sm pxq Cr,m Lipxq. O. °n¥1 pnmµq pϕnpqnmq .. x r m plog xqM. ,. r. Proof. We choose an arbitrary positive constant B, and for every coprime integers a and b, we denote π px; a, bq #tp ¤ x : p a pmod bqu, then Sm pxq. p q πpx; 1, nmq pnmqr n¤x ¸ µ pn q π px; 1, nmq p nmqr n¤plog xq. . ¸ µ n. .

(50). ¸. O. plog xqB n¤x. B. 1. pnmqr πpx; 1, nmq .. The sum in the error term is ¸. plog xqB n¤x. 1 pnmqr πpx; 1, nmq. ¤ ¤. 1 mr. ¸. ¡plog xqB. n. ¸. 1 mr 1 34. 1 nr. ¤¤ p. 2 a x a 1 mod mn. x. ¡plog xq. n. ¸. B. nr 1. 1. q. x ! mr 1plog xqrB. ..

(51) For the main term we apply the Siegel–Walfisz Theorem [34], which states that for every arbitrary positive constants B and C, if a ¤ plog xqB , then . Lipxq ϕpaq. π px; 1, aq . x plog xqC. O. .. So, if we restrict m ¤ plog xqD for any positive constant D, Sm p xq . ¸. ¤plog xqB. n. µ pn q pnmqr ϕpmnq Lipxq . Cr,m Lipxq. O. ¸. ¡plog xqB. n. . Cr,m Lipxq. O. . O. 1 mr ϕpmq. x. mr. . 1. plog xqrB. O. x plog xqC

(52). Lipxq. pnmqr ϕpnmq ¸. ¡plog xqB. n. n. O

(53). n. Thus. ¡plog xqB. 1 r m ϕpmq. x log log x mr plog xqC . Lipxq n r ϕ pn q. 1 pnmqr. O. . O. x log log x mr plog xqC. . O. x r 1 m plog xqrB. x mr 1 plog xqrB. ,. n ϕ pn q ¸. ¤plog xqB . where we have used the elementary inequality ϕpmnq have (see [1, Theorem 8.8.7]). then.

(54). ¸. eγ log log n !. 1 nr ϕpnq. 3 log log n. ¸. ¡plog xqB. n. ¸. 1. ¡plog xqB. n. pn q. nr ϕ. ¥ ϕpmqϕpnq. Since, for every n ¥ 3, we ! log log n ,. log log n nr 1. Lipxq !. (3.9). x ! logploglogxlog qrB .. mr ϕ. x pmqplog xqrB ,. proving the lemma for a suitable choice of D, B and C. The following Lemma concerns the Titchmarsh Divisor Problem [33] in the case of primes p 1 pmod mq. Asymptotic results on this topic can be found in [4] and [5]. Lemma 3.3.4. Let τ be the divisor function and m positive constant D, we have the following inequality: . ¸. ¤. p x p 1 mod m. p. τ. q. p1 m. 35. P. N. If m. ¤ 8x . m. ¤ plog xqD. for an arbitrary.

(55) Proof. Let us write p 1 mjk so that jk the three cases. ¤ px 1q{m and let us set Q . b. and distinguish. x 1 m. ¤ Q, k ¡ Q, j ¡ Q, k ¤ Q, j ¤ Q, k ¤ Q.. • j • •. So we have the identity . ¸. τ. ¤. p x p 1 mod m. p. q. p1 m. . ¸. ¸. ¤. ¤. j Q. . 2. 1 Q2 j. ¸. ¤. ¸. ¤. ¤. k Q. Q k mjk 1 prime. ¸. ¸. ¤. . 2. ¸. ¤. p. ¸. . ¤. π pmkQ. ¤. ¤. 1. k Q mjk 1 prime. 1. ¤ p. p mkQ 1 p 1 mod km. q. 1; 1, kmqq. 1; 1, kmq. k Q. 2. ¸. pπpx; 1, kmq πpmkQ. k Q. ¸. k Q. q. j Q. Q j mjk 1 prime. ¤. k Q mkQ 1 p x p 1 mod km. ¸. ¤. Q2 k. ¸. 1. ¸. 1. π px; 1, kmq . k Q. ¸. ¤. π pmkQ. 1; 1, kmq .. k Q. Using the Montgomery–Vaughan version of the Brun–Titchmarsh Theorem: π px; a, q q ¤. 2x , ϕpq q logpx{q q. for m ¤ plog xqD with D arbitrary positive constant, then we obtain . ¸. ¤. p x p 1 mod m. p. τ. q. p1 m. ¸. ¤. 2. ¤. ¸ 8x 1 . logpx{mq k¤Q ϕpkmq. 2x ϕ p km q log px{kmq k ¤Q. ¤ logpx4x{mQq. ¸. 1 ϕ p km q k ¤Q. Now, substitute the elementary inequality ϕpkmq ¥ mϕpk q and use a result of Montgomery [19] ¸. 1 ϕ p kq k ¤Q. A log Q 36. . B. O. log Q Q. ,.

(56) where A. ζ p2 qζ p3 q ζ p6 q. 1.9436 . . .. 8 µ2 n log n ¸. Aγ . and B. pq 0.0606 . . . , nϕpnq. . n 1. which in particular implies that, for Q large enough, A log Q 1 ¤ Finally. ¸. 1 ϕ p kq k ¤Q . ¸. ¤. τ. p x p 1 mod m. p. q. ¤ A log Q ¤ logpx{mq . p1 m. ¤ 8x . m. Lemma 3.3.5. Let p be an odd prime number and let χ χ0 be a non-principal Dirichlet character modulo p. Define ¸ |cmpχq| , dm,i pχq : r x χP Fp. . χi χ. then. 1 ¹ 1 m p1 `|. dm,i pχq ¤. 1 `. .. m. Proof. From equation (3.6) and Lemma 3.3.1, we have dm,i pχq . 1. pp 1q. . ¸ r. . P ppZ1qZ ni n. n.

(57). p. ¸. 1. p p 1q. . pp 1q{m . , µ2 p1 r p p1q{m , n Jr m p1. where χ e2πin{pp1q with n P Z{pp 1qZzt0u; naming t dm,i pχq . . r. Jr. |. . µ. dt. 2. p 1 m. m. ,n. q. and u gcd pt, ni q we get. p 1 m. Jr ptq H pdq , Jr dt. t d. where #. H pdq : # x P. . Z pp 1 qZ. r 1. +. : pu, xq d. 37. . . p1 d. r1. p q.. ¸µ k k. | ud. . kr 1.

(58) Then dm,i pχq . ¸. 1. pp 1q d|t. ¤ p 1 1. 3.4. ¸. |. . µ. t d. 2. . µ2. dt. t d. ¸ µp k q Jr ptq dr1 Jr dt k| u k r1 d. d. t. p1. p q 1 k m. ¸ µ2 k. |. kt. ¹ `. | pm1. . 1. 1 `. Proof of Theorem 1. We follow the method of Stephens [31]. By exchanging the order of summation we obtain that ¸. Nxay,m pxq . P ¤. Zr. a 0 a1 T1. ¸ p x p 1 mod m. p. .. .. ¤. Mpm pT q ,. q. ¤. 0 ar Tr. where Mpm pT q is the number of r-tuples a P Zr , with 0 ai ¤ Ti and vp pai q i 1, . . . , r, whose reductions modulo p satisfies rFp : xayp s m. We can write Mpm pT q . ¸. P. a Zr 0 a1 T1. 0 for each. tp,m paq ,. ¤ .. .. ¤. 0 ar Tr. with tp,m paq . #. 1 if rFp : xayp s m , 0 otherwise .. Given a r-tuple χ of Dirichlet characters mod p, by orthogonality relations it is easy to verify that ¸ tp,m paq cm pχqχpaq ; (3.10) p qr χPpFx so we have. ¸. P. a Zr 0 a1 T1. ¤ .. .. Nxay,m pxq . ¸. ¸. ¸. ¤. P. Pp q. p x a Zr χ Fx p p 1 mod m 0 a1 T1. p. q ¤ .. .. ¤. ¤. 0 ar Tr. 0 ar Tr. 38. cm pχqχpaq . r. (3.11).

(59) Let χ0 : pχ0 , . . . , χ0 q be the r-tuple consisting of all principal characters, then cm pχ0 q. pp 1 1qr. ¸ a F p Fp : a p. Pp qr r x y sm. χ0 paq. pp 1 1qr #ta P pZ{pp 1qZqr : pa, p 1q mu pp 1 1qr Rppmq . ±. Denoting |T | : ri1 Ti and T : mintTi : i 1, . . . , ru, through (3.8) and (3.7), we can write the main term in (3.11) as 1 |T |. ¸. ¸. ¤. P. p x a Zr p 1 mod m 0 a1 T1. p. cm pχ0 qχ0 paq. q ¤ .. .. ¤. 0 ar Tr. |T1 |. ¸. ¤. p x p 1 mod m. p. ¸. . cm pχ0 q. q. q. p x p 1 mod m. p. . Sm pxq. 1 |T |. ¸. ¤. p x p 1 mod m. p. Oplog log xq. . . O. i 1. 1 Ti. . O. x T log x. q. 1 p. . O. x T log x. .. ¡ 0, and T ¡ expp4plog x log log xq1{2q, so we can apply Lemma. ¸. ¸. ¤. P. p x a Zr p 1 mod m 0 a1 T1. p. r ¸. 1 pr.

(60). O . q. By hypothesis m ¤ plog xqD , D 3.3.3 to obtain. . r p. . cm pχ0 q. ¤. ttTiu tTi{puu. . cm pχ0 q 1 . ¤. ¸. . . i 1. p x p 1 mod m. p. r ¹. cm pχ0 qχ0 paq Cr,m Lipxq. q ¤ .. .. ¤. 0 ar Tr. 39. . O. x mr plog xqM. ,.

(61) where M. ¡ 1. For the error term we need to estimate the sum. Er,m pxq :. ¸ ¸ cm χ . r p ¤x p ztχ u χP Fx 0 pmod mq . p q. 1. |T |. . p 1. !. r ¸ 1. Ti. i 1. ¸ χa aPZr 0 a1 ¤T1 .. . 0 ar ¤Tr ¸ χ χa , aPZ . ¸. ¸. ¤. q P zt u. p x p χ0 χ Fx p 1 mod m. p. dm,i p. pq. q. (3.12). pq. ¤. 0 a Ti. since the r main contributions to (3.12) comes from the cases in which just one Dirichlet character in χ is non-principal, say χi χ χ0 , while for every j i we choose χj χ0 , giving ¸ χa aPZr 0 a1 ¤T1 .. . . pq. ¸ χa aPZ 0 a¤Ti . pq. ¸ ¸ 1 0¤j ¤r aj PZ j i 0 aj ¤Tj p-a. T ¸ χa . Ti aPZ . ¤| |. pq. ¤. 0 a Ti. j. ¤. 0 ar Tr. Define i Er,m pxq :. ¸. ¸. ¤. q P zt u. p x p χ0 χ Fx p 1 mod m. p. dm,i p. ¸ χ χa , aPZ . q. pq. (3.13). ¤. 0 a Ti. then by Holder’s inequality. i Er,m px q. (2si. $ ' ' &. ¤' ' %. . ¸. ¸. ¤. q P zt u. p x p χ0 χ Fx p 1 mod m. p. ¸. ¤. 2si 2si 1. . q. / / -. 2si ¸ ¸ χ a . p ztχ0 u aPZ χPFx. pq. p x p 1 mod m. p. tdm,ipχqu. ,2si 1 / / .. 40. ¤. 0 a Ti. (3.14).

(62) As before, given a primitive root g modulo p, write χj pg q with nj P Z{pp 1qZ, so that by equation (3.6). ¸ . p r ztχ u χP Fx 0. cm pχq . j. ¸. 1. pp 1 q. e2πin {pp1q for every j 1, . . . , r,. r. . P pq. n. Z p 1 Z. r. zt0u. c p1 pnq . m. Denoting again t pp 1q{m, from Lemma 3.3.1 we derive the following upper bound:. ¸. p ztχ0 u χPFx. dm,i pχq. ¤. ¸ . p χP Fx. ¤. ¸. . ¸. . ¹. |. r. µ2. dt. |. dt. |. ztχ0 u. . t d. . µ2 . t d. 1. `t. Call Dm,i ppq : maxtdm,i pχq : χ. |cmpχq| . Jr ptq r pp 1qr Jr pt{dq # tn P pZ{pp 1qZq : pt, nq du. Jr ptq ¸ µpk q dr Jr pt{dq t k r. |d 1 2ωptq ¤ 2ωptq .. k. Jrtprtq. ¸. |. dt. . µ2. t d. `r. P Fxp ztχ0uu, then the following asymptotic estimate holds for 41.

(63) every si. ¥ 1: ¸. ¸. ¤. q P zt u. p x p χ0 χ Fx p 1 mod m. p. tdm,ipχqu. 2si 2si 1. . ¤. ¸. ¸. ¤. q P zt u. 1. p x p χ0 χ Fx p 1 mod m. p. ¸. ¤. ¤. dm,i pχqtdm,i pχqu 2si 1. tDm,ippqu. 1 2si 1. tDm,ippqu. 1 2si 1. p ztχ0 u χPFx. p x p 1 mod m. p. q. ¸. ¤. ¤ p q 1 ¤ m 2si1. ¹. ¤. p1 q |m. p x ` p 1 mod m. p. ¸. 1 ! m 2si1. ¹. ¤. p1 q |m. . 1 `. 1 . 1. p x ` p 1 mod m. p. ¤. τ. p x p 1 mod m. ! m. p. 1 ` . ¸. 1 ! m 2si1 log log x. dm,i pχq. 1 q 2ωp pm. p x p 1 mod m. ¸. ¸. . q. 2ωp. 1. q. p 1 m. 2ωp. p1 m. q. p 1 m. 2si 2si 1. x log log x ,. where we have used Lemma 3.3.4 and Lemma 3.3.5 together with the simple observation. tmDm,ippqu. . 1 2si 1. ¹. ¤ `. . | pm1. 1 `. 1. . 1 2si 1. ¤. ¹. | pm1. `. . 1. 1 `. .. To estimate the other factor in (3.14) we use Lemma 5 in [31]:. ¸. ¤. p x p 1 mod m. p. 2si ¸ ¸ χ a a P Z x χPFp ztχ0 u. pq. q. ! px2. ¤. 0 a Ti. 42. Ti si qTi si plogpeTi si 1 qqsi. 2. 1 ..