Topics in analytic number theory

(1)

Topics In Analytic Number Theory

Thèse

Patrick Letendre

Doctorat en mathématiques

Philosophiæ doctor (Ph. D.)

Québec, Canada

(2)

Topics in Analytic Number Theory

Thèse

Patrick Letendre

Sous la direction de:

(3)

Résumé

Le présent document est un compte-rendu de quatre présentations que j'ai faites au congrès de Théorie des Nombres Québec-Maine entre 2013 et 2016. Au l des ans, j'ai eectué quelques améliorations et corrections aux documents originaux. Le contenu, l'esprit et l'organisation sont restés essentiellement inchangés. Les quatre sujets sont fondamentalement distincts tout en étant dans un même cercle d'idées.

Le premier chapitre traite d'un certain nombre de sujets en relation avec le comporte-ment moyen de certaines fonctions multiplicatives, dans un ensemble bien précis, qui partagent plusieurs propriétés avec la fonction indicatrice des nombres libres de puis-sance k-ième. En particulier, on y établit plusieurs estimations de la variance dans des intervalles courts et dans des progressions arithmétiques.

Le deuxième chapitre étudie un problème du crible combinatoire. Il y est question d'établir une majoration analogue à la célèbre inégalité de Brun-Titchmarsh, mais pour les nombres libres de puissance k-ième. Après quelques remarques élémentaires, on établit une nouvelle inégalité en supposant une conjecture forte en lien avec la densité maximale d'une suite de nombres ayant un diviseur de la forme pk

1pk2 où p1 et p2 sont

des nombres premiers qui satisfont certaines conditions. La méthode fournit aussi une majoration eective pour le nombre de nombres libres de puissance k-ième dans un intervalle [x + 1, x + h] lorsque h est petit par rapport à x.

Le troisième chapitre, écrit en collaboration avec Jean-Marie De Koninck, établit des inégalités particulières pour la fonction τ(n) qui compte le nombre de diviseurs de n. L'objectif est d'obtenir une majoration de τ(n) qui ne dépend pas des facteurs premiers de n, mais seulement du nombre de facteurs premiers distincts de n et de son ordre de grandeur, i.e. de log n. L'inégalité principale (Théorèmes3.4 et3.5) a nécessité un bon volume de calcul sur ordinateur, et donc beaucoup de programmation avec Maple. Finalement, le Chapitre 4 est le début d'une étude du nombre de points entiers près

(4)

d'une courbe dans l'espace R3_{. Le problème peut aussi être vu comme celui du nombre}

de points entiers près de deux courbes dans le plan Euclidien simultanément. L'objectif principal est d'utiliser l'information des deux courbes de façon nontriviale, soit de faire mieux que les meilleurs résultats connus pour une seule courbe. Étant donné la com-plexité du problème déjà en deux dimensions et du nombre de méthodes disponibles, il nous a semblé impossible de faire un traitement complet de la question. On s'est donc concentré sur une méthode qui utilise des approximations linéaires. Cette dernière peut sans doute être substantiellement améliorée.

(5)

3.2 Background results . . . 92 3.3 Main results . . . 95 3.4 Preliminary lemmas. . . 96 3.5 Proof of Theorem 3.1 . . . 106 3.6 Proof of Theorem 3.2 . . . 108 3.7 Proof of Theorem 3.3 . . . 110 3.8 Proof of Theorem 3.4 . . . 111 3.9 Proof of Theorem 3.5 . . . 116 3.9.1 Preliminary steps . . . 116 3.9.2 A rst argumentation . . . 117 3.9.3 A rst verication. . . 119

3.9.4 Reducing the upper bound for δ0 _{. . . .} ₁₂₀

3.9.5 The last verication . . . 123

3.10 Final remarks . . . 124

Bibliography 126 4 Lattice points close to a three-dimensional smooth curve 128 Résumé 128 Abstract 128 4.1 Introduction and notation . . . 129

4.2 Statement of the Theorems. . . 131

4.3 Preparatory lemmas . . . 133

4.3.1 Minor arcs . . . 135

4.3.2 Relationship between f and a general plane . . . 136

4.4 Linear major arcs . . . 139

4.5 Major arcs . . . 152 4.6 Proof of Theorem 4.1 . . . 158 4.6.1 Setting a structure on S . . . 159 4.6.2 The S2 term . . . 161 4.6.2.1 The case s = 1 . . . 163 4.6.2.2 The case s ≥ 2 . . . 165

(7)

4.6.2.3 Another method . . . 168

4.6.3 The S1 term . . . 172

4.6.4 The exceptional cases . . . 179

4.9 Examples and concluding remarks . . . 182

Bibliography 184

(8)

Remerciements

Je tiens d'abord à remercier l'université Laval pour m'avoir fourni un excellent en-vironnement d'étude pendant plusieurs années. Ensuite, je remercie l'ensemble des professeurs du département de mathématiques et de statistique qui ont participé à mon développement en tant que mathématiciens et en tant qu'humains. Des remer-ciements bien particuliers pour monsieur Claude Levesque qui a été mon directeur pendant plusieurs années, notamment à la maîtrise. Je vais me rapeller de plusieurs discussions sur diérents sujets de théorie des nombres. Et enn, je remercie mon directeur monsieur Jean-Marie De Koninck, le célèbre mathématicien aux 1001 occu-pations, pour sa façon naturelle de communiquer sa passion pour les mathématiques et pour d'innombrables conseils de toutes sortes.

Je remercie mes véricateurs, i.e. Jean-Marie De Koninck, Nicolas Doyon, Sary Drap-peau et Antonio Lei. Chacun d'eux m'a fourni une liste de suggestions qui m'ont aidé à améliorer fortement la qualité du document.

Je remercie tous ceux qui ont contribué de près ou de loin à une idée ou chose qui m'alimente encore aujourd'hui d'une quelconque façon.

Je remercie ma famille extraordinaire qui m'encourage depuis toujours, avec une pensée très spéciale qui va à mes amis, ma famille rapprochée et la famille de mon frère. Pour nir, je remercie ma mère pour tout.

(9)

Avant-propos

Les 4 chapitres de cette thèse contiennent des résultats à publier sous une forme ou une autre. Les Chapitres 1, 2et4ont été entièrement conçus et écrits par l'auteur de cette thèse. Le Chapitre 3 est un article écrit en collaboration avec Jean-Marie De Koninck. La contribution de ce dernier est une preuve simpliée du Lemme 3.5 et une grande quantité de commentaires et conseils judicieux. Aucun des chapitres n'a été soumis en date du dépôt nal de cette thèse.

(10)

Introduction

In Chapter 1, we consider the class of multiplicative functions f that can be written as f (n) =X

dk_|n

g(d)

where k ≥ 2 is xed and g is a multiplicative function for which we require that |g(p)| = 1 and |g(pj_{)| ≤ 1} _{for j ≥ 2.}

Such a function f is said to be in the class Ck.

One can show that for each such function f we have X n≤x f (n) = cfx + Ef(x) with |Ef(x)| x1/k (x → ∞) where cf := X n≥1 g(n) nk .

We want to estimate how the average of a function f ∈ Ck behave in most intervals.

For this reason, we evaluate M(h, N ) := N −1 X n=0 h X a=1 f (n + a) − cfh 2 (h, N ∈ N) as well as T (h, M ; β) := M X n=0 h X a=1 f (nh + a + β) − cfh 2 (β = 0, . . . , h − 1). Here, h, M and N are integers that satisfy h(M + 1) = N.

Another similar topic is the study of how the average of such a xed function f behave in arithmetic progressions. Again, given an integer q ≥ 2, one can prove the estimate

Q(x; q, a, f ) := X

n≤x n≡a mod q

(11)

where g(q, a, f ) := 1 q Y p-q X i≥0 g(pi₎ pik Y p|q X i≥0 (pik_,q)|a g(pi_)(pik_{, q)} pik .

We are thus led to consider the quantity W(x; q, f ) :=

q

X

a=1

|E(x; q, a, f )|2_.

In Chapter 2, we denote by µk the indicator function of the set of k-free numbers

(k ≥ 2). For any positive integers x and h, let Qk(x) := x X n=1 µk(n), Gk(x, h) := Qk(x + h) − Qk(x) and gk(h) := max x≥0 Gk(x, h). We are interested by gk(h) as h → ∞.

In Chapter 3, we let τ(n) stand for the number of divisors of the positive integer n and ω(n) stand for the number of prime factors of the positive integer n. We shall also be using the functions

γ(n) :=Y p|n p and b(n) :=Y p|n 1 log p.

In 1915, Ramanujan obtained the inequality τ (n) ≤ log(nγ(n))

ω(n)

b(n) (n ≥ 2).

We want to nd an upper bound for the function τ(n) that does just depend on ω(n) and the size of n, that is log n.

In Chapter 4, we let X := [X, 2X] be a xed interval, with X ≥ 2. We write f(x) := (x, f1(x), f2(x)), where f1 and f2 are xed functions in Cn(X ) for some n ≥ 1. Much

(12)

work has been done to count the number of integer points close or on a given curve in R2_{. In this paper, we want to estimate the number of integer solutions close to the}

curve f(x) when x ∈ X . That is, if we write

Λ := {(v1, v2, v3) ∈ R3| ∃x ∈ X s.t. v1 = x, |v2− f1(x)| < δ1 and |v3− f2(x)| < δ2},

then we want to give good upper bounds for S := #(Λ ∩ Z3₎_{. It is assumed that}

0 ≤ δ1, δ2 ≤ 1/4. We are ready to assume various conditions on derivatives of the

(13)

Chapter 1 On a class of multiplicative functions

Patrick Letendre

Résumé

Pour chaque fonction f dans un ensemble de fonctions multiplicatives précisé-ment dénies, nous estimons la variance des sommes prises dans les intervalles de longueur h et dans les progressions arithmétiques de raison q.

Abstract

For each function f in a set of precisely dened multiplicative functions, we es-timate the variance of the sums taken in the intervals of length h and in the arithmetic progressions of common dierence q.

(14)

1.1 Introduction

For each k ≥ 2, we note by Ck the class of multiplicative functions f that are supported

on k-th powers, that is

(1.1.1) f (n) =X

dk_|n

g(d)

for some multiplicative function g for which we require that |g(p)| = 1 and |g(pj_{)| ≤ 1} _{for j ≥ 2.}

This class contains the indicator function of the k-free numbers, noted µk, that satises

(1.1.2) µk(n) :=

X

dk_|n

µ(d) (k ≥ 2)

and also the functions τ1,k(n) dened more generally by

(1.1.3) τj,k(n) :=

X

dj_ek_=n

1 _{(j, k ∈ N).}

A large source of examples is provided by the choice of any additive function t : N → R. Indeed, in this case we can consider the function f that is dened by g(d) = exp(2πit(d)) in (1.1.1).

The asymptotic behavior of functions in this class is well known. In fact, one can easily show that for each such function f we have

(1.1.4) X n≤x f (n) = cfx + Ef(x) with |Ef(x)| x1/k (x → ∞) where (1.1.5) cf := X n≥1 g(n) nk .

It turns out that the class Ck has received a great deal of attention in the literature,

often through one of the functions dened in (1.1.2) and (1.1.3). Here, we focus on some estimates that we believe to be of some interest for the distribution of the values of a xed function f ∈ Ck in small intervals and in arithmetic progressions. All the

techniques are elementary, while for certain particular functions, analytic methods will provide better results.

(15)

One of the rst thing that comes to mind, for a given f ∈ Ck, is the variance (1.1.6) M(h, N ) := N −1 X n=0 h X a=1 f (n + a) − cfh 2 (h, N ∈ N)

and the same with a higher power. Hall in [9] has evaluated (1.1.6) in the case where the function f is the indicator function of the squarefree integers. As we will see, the quantity (1.1.7) T (h, M ; β) := M X n=0 h X a=1 f (nh + a + β) − cfh 2 (β = 0, . . . , h − 1) is related to the variance M(h, N). Here, h, M and N are integers that satisfy h(M + 1) = N, in which case we clearly have the identity

(1.1.8)

h−1

X

β=0

T (h, M ; β) = M(h, N )

that will be used in the process of the evaluation of (1.1.7), namely in Theorem1.2. We also provide an estimate for a more general quantity with higher powers. The result is not of the same quality but leads to better results to the question of how many intervals have a large error term.

Another similar situation is when the partitioning of the interval [1, N] is done with arithmetic progressions. Again, given an integer q ≥ 2, one can easily prove the estimate (1.1.9) Q(x; q, a, f ) := X n≤x n≡a mod q f (n) := g(q, a, f )x + E(x; q, a, f ) where (1.1.10) g(q, a, f ) := 1 q Y p-q X i≥0 g(pi) pik Y p|q X i≥0 (pik_,q)|a g(pi)(pik, q) pik .

We are thus led to consider the quantity (1.1.11) W(x; q, f ) :=

q

X

a=1

|E(x; q, a, f )|2.

An asymptotic result for a similar quantity is the main theme in a paper of Nunes [15]. There are many papers that study the distribution in arithmetic progressions with an extra average over the values of q ≤ Q. Some examples of this are in Warlimont [20],

(16)

Vaughan [19] and in the paper of Brüdern, Granville, Perelli, Vaughan, Wooley [1]. The strongest general result that we are aware of is

(1.1.12) W(x; q, f ) (2k)ω(q)log4x x1/kq1−1/k+x 2/k q1/k

where ω(q) stands for the number of distinct prime factors of q. One can easily obtain (1.1.12) with the methods used in the paper of Le Boudec [14]. In Theorem 1.3 we obtain an asymptotic expression for W(x; q, f) by a combination of methods from the papers [13], [2], [3] and the method that leads to Theorem1.2.

We also present a series of estimates inspired by the famous Bombieri-Vinogradov theorem for the distribution of primes in arithmetic progressions. We believe that this idea rst originated from Orr in his thesis [16]. Here we present a version valid in a small interval [x + 1, x + y] and obtain results in terms of the quantity

(1.1.13) Sk(x, y) := #{d > y1/k| χ[x+1,x+y](d) = 1}

where χ[x+1,x+y] is the indicator function of the integers d for which there exists an

integer m such that mdk _{∈ [x + 1, x + y]}. We present in Lemma 1.2 the simplest

nontrivial estimate of Sk(x, y).

1.2 Statement of the theorems

Fix an integer k ≥ 2. Given h and M in N, write (1.2.1) γf := Y p 1 − 1 p X a≥0 1 pa |g(p a_)|2_{+ 2<}X i≥1 g(pa_)g(pa+i₎ pik !! , (1.2.2) `(ξ) :=    log 2h if ξ = 1/2, 1 otherwise and (1.2.3) `1(ξ) :=    log 2M if ξ = 1/2, 1 otherwise.

Theorem 1.1. Let f be a function in Ck. Then there exists a positive constant θk ≤ _k+21

such that M(h, N ) = 2ζ(1/k − 1) 1/k − 1 γfh 1/k_{N + O N h}1/2k_{+ N h}θk_`(kθ k) +O N2/(k+1)h2−2/(k+1)log N + h2N1/klog N.

(17)

In particular we have

M(h, N ) = (1 + o(1))2ζ(1/k − 1) 1/k − 1 γfh

1/k_N

in the region 1 ≤ h ≤ H where H = o N(k−1)/(2k−1) logk/(2k−1)N

as H, N → ∞.

Theorem 1.2. Let f be a function in Ck. Then for each β ∈ {0, . . . , h − 1} there exists

a positive constant θk ≤ _k+21 such that

T (h, M ; β) = 2ζ(1/k − 1) 1/k − 1 γfh 1/k_{(M + 1) + O} M hθk_`(kθ k) +O M h1/2kY p|h 1 + 1 p1/2 + 1 p Y pj_kh (r−1)k<j≤rk 2r − 1 + 2 p j prk +O h2M1/klog N + hM2/(k+1)log NY p|h 1 + 2 pk/(k+1) . In particular we have T (h, M ; β) = (1 + o(1))2ζ(1/k − 1) 1/k − 1 γfh 1/k (M + 1)

in the region 1 ≤ h ≤ H where H = o N(k−1)/(3k−2) logk/(3k−2)N

as H, N → ∞.

Theorem 1.3. Let f be a function in Ck. Assume that x = qM where M is a real

number with M ≥ 1. Then there exists a positive constant θk ≤ _k+21 such that

W(x; q, f ) = 2ζ(1/k − 1) 1/k − 1 γf(q)qM 1/k_{+ R(x; q, f ),} where γf(q) := Y p-q 1 −1 p X j≥0 1 pj |g(pj_)|2_{+ 2<g(p}j₎X i≥1 g(pj+i) pik × Y pα_kq (r−1)k<α≤rk 1 −1 p r−1 X j=0 1 pjk |g(p j_)|2_{+ 2<¯}_g(pj₎X i≥1 g(pj+i₎ pik ! + 1 pα−α/k X i≥r 1 pi |g(p i )|2 + 2<¯g(pi)X l≥1 g(pi+l) plk !!

(18)

and R(x; q, f ) 2ω(q)qY p|q 1 + 1 p + M2k1 ` 1(kθk)q Y p|q 1 + 1 p1/2 + 1 p +Mθk_qY p|q 1 + 1 pkθk +2 ω(q)_x1+1/k_{log x} q .

1.3 Some notation

The subscript (k) to a sum X

(k) t|q

means that it runs over the divisors t of q = pα1

1 · · · p αj

j

which are of the form t = pβ1

1 · · · p βj

j where each βi ∈ {0, k, 2k, . . . , bαi/kck, αi}. The

superscript 0 _{to a sum} X0

t|q

means that the sum runs over the squarefree divisors t of q. A double sum like Ph_a=1Ph

b=1 is simply written as P h a,b=1.

For a xed value of k, we consider the function

(1.3.1) η(n) := Y

pα_kn

pdαke.

It has the nice property that if (dk_{, q) = r} _{then η(r) | d.}

By m ∼ M we mean that m ∈ [M, 2M).

1.4 Preparatory Lemmas

Lemma 1.1. Let (an) be a sequence of complex numbers. Then the identity

(1.4.1) N X n=1 |an− T |2 = N X n=1 |an|2− 1 N N X n=1 an 2 + N ||2, holds with T := 1 N N X n=1 an+ with ∈ C.

Proof. By expanding the LHS of (1.4.1), we nd

N X n=1 |an|2− 2< T N X n=1 an ! + N |T |2. The result then follows by using the denition of T .

(19)

Lemma 1.2. For each k ≥ 2, we have Sk(x, h) := X d>h1/k X x<n≤x+h dk_|n 1 h1/k+ xk+21 (x, h ≥ 1).

Proof. We use the rst and second derivative estimate for the number of integers close to a smooth curve as explained in the paper of Huxley and Sargos [11]. We consider the function r(d) := x dk, so that we have λ1 := x Mk+1, λ2 := x Mk+2 and δ := h Mk

for d ∈ (M, 2M]. Now, the rst test is

λ1M + δM + δ/λ1+ 1 = (λ1M + 1)(δ/λ1+ 1).

Such an inequality is easily obtained but not explicitly written in the cited paper. We have that X M <d≤2M X x<n≤x+h dk_|n 1 is bounded both by x Mk + h Mk−1 + hM x + 1 and by x1/3 Mk−13 + h Mk−1 + h1/2M x1/2 + 1.

Using these estimates, we nd that Sk(x, h) blog((x+h)/h)/ log(2)c X j=0 h (2j_h1/k₎k−1 + min 2 j_h1/k_, x (2j_h1/k₎k, x1/3 (2j_h1/k₎k−1₃ ! h1/k+ xk+21 ,

while the other contributions are negligible or included in this estimate.

Lemma 1.3. Let a1, . . . , am and n1, . . . , nm be respectively a sequence of integers and

of positive integers. The number of solutions (mod [n1, . . . , nm]) to the system of

congruences

n ≡ ai (mod ni) i = 1, . . . , m

is

(1.4.2) ( 1 if (ni, nj)|(ai − aj) for i 6= j,

(20)

Corollary 1.1. Let h ≥ 1 be an integer and let a1, . . . , am and n1, . . . , nm be

re-spectively a sequence of integers and of positive integers. The number of solutions (mod [h,n1,...,nm]

h ) to the system of congruences

hn ≡ ai (mod ni) (i = 1, . . . , m)

is

( 1 if (ni, nj)|(ai− aj) for 1 ≤ i < j ≤ m and (h, ni)|ai for 1 ≤ i ≤ m,

0 otherwise.

Proof. We apply Lemma 1.3 to the system of congruences t ≡ ai (mod ni) for 1 ≤ i ≤ m

t ≡ 0 (mod h).

Lemma 1.4. Given a set {n1, . . . , nm}, consider the functions

Sj := Sj(n1, . . . , nm) =

Y

1≤i1<···<ij≤m

(ni1, . . . , nij) (j = 1, . . . , m).

Then, we have the identity

[n1, . . . , nm] = m Y j=1 S_j(−1)j+1. Lemma 1.5. Let h ≥ 1 be an integer. Then the identity

h X a,b=1 t|a−b 1 = h 2 t + tP h t holds with P (x) := {x} − {x}2 (here {x} stands for the fractional part of x).

(21)

Lemma 1.6. For k ≥ 2, we have (1.4.3) ∞ X d=1 P M dk = 2ζ( 1 k − 1) 1 k− 1 M1/k+ O(Mθk_), where θk≤            1 k+2 for k ∈ {2, 3, 4, 5, 6}, 6 7k+6 for k ∈ {7, 8, 9, 10, 11}, 1 k+3 for k ≥ 12.

Proof. Firstly, we have X d>M1/k P M dk = X d>M1/k M dk − M2 d2k = M 1/k 1 k − 1 − 1 2k − 1 + O(1), where we used the Euler-Maclaurin formula to show that

N X n=1 1 nr = ζ(r) + N1−r 1 − r + 1 2Nr + O r Nr+1

and also that

ζ(s) = 1 2 + 1 s − 1 + s(s + 1) 2 Z ∞ 1 P (x) xs+2dx

holds in the region < s > −1. For the second part, we write

bM1/k_c X d=1 P M dk = Z M1/k 1 P M tk dt − [M1/k_] X n=1 ψ2 M nk − Z n n−1 ψ2 M tk dt + O(1), where ψ2(x) := B2({x}) is the second Bernoulli function

ψ2(x) = {x}2− {x} + 1/6 = 1 2π2 X n∈Z\{0} e(nx) n2

(here e(y) = e2πiy_{) that possesses an expansion as an absolutely convergent Fourier}

series. We evaluate the rst integral by doing the change of variable x = M

tk, and obtain Z M1/k 1 P M tk dt = M1/k k Z M 1 P (x) x1+1/kdx = M1/k k Z ∞ 1 P (x) x1+1/kdx + O(1).

To evaluate the sum, we begin by splitting it into parts with a parameter N to be chosen later. For the large values of n, we use the fact that ψ2 is continuous and satises

max x1,x2∈[n,n+1] ψ2 M xk 1 − ψ2 M xk 2 kM nk+1

(22)

and nd that X N <n≤M1/k ψ2 M nk − Z n n−1 ψ2 M tk dt M Nk.

Then, we estimate the integral by using the rst derivative test (see [12] Lemma 8.10) on each term of the Fourier series

Z N 1 ψ2 M tk dt = 1 2π2 X n6=0 1 n2 Z N 1 e nM tk dt X n6=0 1 n2 1 knM + Nk+1 knM N k+1 kM . Finally, we use Theorem 2.2 of [6] to get

2T X n=T ψ2 M nk = 1 2π2 X n6=0 1 n2 2T X m=T e nM mk X n6=0 1 n2 T k2_nM Tk+2 1₂ + T k+2 k2_nM 12! kM 1/2 Tk/2 + Tk/2+1 kM1/2

and nd the inequality

N X n=1 ψ2 M nk W + [log(N/W )/ log(2)]+1 X j=0 kM1/2 (2j_{W )}k/2 + (2j_{W )}k/2+1 kM1/2 W +kM 1/2 Wk/2 + Nk/2+1 kM1/2 M 1 k+2 + N k/2+1 kM1/2

holds with W := (kM1/2₎2/(k+2)_{. So far we have showed that} ∞ X d=1 P M dk = 2ζ( 1 k − 1) 1 k− 1 M1/k+ O Mk+21 + N k/2+1 kM1/2 + Nk+1 kM + M Nk

after some simplications. The result follows with the choice N = (kM2₎2k+11 for k ≥ 4

and N = M 3

3k+2 for k = 2, 3. The other results are obtained in the same manner this

time using Theorem 6.9 of [18]. Observe that better results can be obtained with the theory of exponent pairs.

We now dene the constants (1.4.4) γf(a, b) := ∞ X d,e=1 (dk,ek)|a−b g(d)g(e) [dk_{, e}k_] (a, b ∈ {1, . . . , h}).

(23)

Lemma 1.7. Let f be a function in Ck. Then, with cf, γf and θk respectively dened

in (1.1.5), (1.2.1) and Lemma 1.6, we have F (h) := h X a,b=1 γf(a, b) = |cf|2h2+ 2 ζ(1/k − 1) 1/k − 1 γfh 1/k_{+ O(h}1/2k_{+ h}θk₎ _{(h ∈ N).}

On the other hand, if θk = _2k1 then the error term becomes hθklog h(except if f satises

the additional condition |g(p2_{)| = 1}, in which case it is h1/2k).

Proof. We have F (h) = h X a,b=1 ∞ X d,e=1 (dk_,ek_)|a−b g(d)g(e)(d k_{, e}k₎ dk_ek = ∞ X d,e=1 g(d)g(e)(d k_{, e}k₎ dk_ek h X a,b=1 (dk_,ek_)|a−b 1 = ∞ X d,e=1 g(d)g(e)(d k_{, e}k₎ dk_ek h2 (dk_{, e}k₎+ (d k_{, e}k_)P h (dk_{, e}k₎ = h2|cf|2+ ∞ X d,e=1 g(d)g(e)(d k_{, e}k₎2 dk_ek P h (dk_{, e}k₎ = h2|cf|2+ F1(h), (1.4.5)

say, where we used Lemma 1.5.

The strategy is to split the arithmetic factor in F1(h)to obtain a linear combination of

the expression evaluated in Lemma 1.6. First observe that since f is in Ck,

1 + 2X i≥1 <(g(pi₎₎ pik 6= 0 and 1 +X i≥1 g(pi) pik 6= 0

for each prime number p. Now, observe that

F1(h) = ∞ X j=1 ∞ X d,e=1 (d,e)=j g(d)g(e) dk_ek j 2k_P h jk = ∞ X j=1 ∞ X d,e=1 (d,e)=1 g(dj)g(ej) dk_ek P h jk

(24)

and consider the function ξ(j) := ∞ X d,e=1 (d,e)=1 g(dj)g(ej) dk_ek (j ∈ N).

We easily show that

ξ(j) = βfθ(j) (j ∈ N), where βf := Y p 1 + 2X i≥1 <(g(pi₎₎ pik ! and θ(j) := Y pa_kj 1 + 2X i≥1 <(g(pi₎₎ pik !−1 |g(pa)|2+ 2X i≥1 <(g(pa_)g(pa+i₎₎ pik ! (j ∈ N). Our task will be to evaluate

F1(h) = X j≥1 ξ(j)P h jk = βf X j≥1 θ(j)P h jk (h ∈ R≥0). By setting ν(pi) := θ(pi) − θ(pi−1) _{(i ∈ N),} we obtain F1(h) = βf X j≥1 X d|j ν(d)P h jk = βf X d≥1 ν(d)X j≥1 P h dk_jk

and from there, in light of Lemma 1.6, X n≥1 P x nk =      2ζ(1/k − 1) 1/k − 1 x 1/k_{+ O(x}θk₎ if x ≥ 1, O(x) otherwise.

Therefore, using the inequality z ≤ z1/k _{for 0 ≤ z ≤ 1, we nd that}

F1(h) = 2βf ζ(1/k − 1) 1/k − 1 h 1/kX d≥1 ν(d) d + O  h1/k X d≥h1/k |ν(d)| d + h θk X d≤h1/k |ν(d)| dkθk   = 2βf ζ(1/k − 1) 1/k − 1 h 1/kX d≥1 ν(d) d + O h 1/k_E 1(h) + hθkE2(h) , (1.4.6)

say. We then use the estimates (1.4.7) ν(p) ≤ 4

pk_{− 3} and ν(p

j_{) ≤} pk+ 3

(25)

to show that

X

d≥1

ν(d) d

converges absolutely and that the sum E1(h) in the error term is h−1/2k and that

hθk_E

2(h) is hθk+ h1/2k if θk6= _2k1 and hθklog(h) otherwise (except if the function

satises the stronger condition |g(p2_{)| = 1}_{, in which case it is h}1/2k_{). Indeed, we use}

the same method for each case: we write d = rs where r is squarefree and s powerful. Clearly, E1(h) X 1≤≤h1/k |ν(r)| r X s≥h1/k_/r |ν(s)| s + X r>h1/k |ν(r)| r 1 h1/2k h1/k X r=1 |ν(r)| r1/2 + 1 h1/k Y p 1 + 4 pk_{− 3} 1 h1/2k (1.4.8)

and similarly for E2(h). Using this in (1.4.6), the result follows from (1.4.5).

For a xed β, we dene the constants (1.4.9) γf(h, a, b, β) := X d,e≥1 (h,dk_)|a+β (h,ek_)|b+β (dk_,ek_)|a−b g(d)g(e)h [h, dk_{, e}k_].

Lemma 1.8. Let f be a function in Ck and consider

(1.4.10) G(h) := sup (β1,β2)∈{0,...,h−1}2 h X a,b=1 γf(h, a, b, β1) − h X a,b=1 γf(h, a, b, β2) (h ∈ N). Then, G(h) h1/2kY p|h 1 + 1 p1/2 + 1 p Y pjkh (r−1)k<j≤rk 2r − 1 + 2p j prk . Proof. We rst write h X a,b=1 γf(h, a, b, β) = X d,e≥1 hg(d)g(e) [h, dk_{, e}k_] h X a,b=1 (dk_,ek_)|a−b (h,dk)|a+β (h,ek)|b+β 1.

(26)

Using Lemma 1.4 to evaluate [h, dk_{, e}k_]_{, we nd that} G(h) ≤ X (k) w|h X (k) r,s|h (r,s)=w X t≥1 (tk_,h)=w X d,e≥1 (d,e)=t (h,dk)=r (h,ek_)=s rstk dk_ek_w h X a,b=1 tk_|a−b r|a+β1 s|b+β1 1 − h X a,b=1 tk_|a−b r|a+β2 s|b+β2 1 .

We easily show that

(1.4.11) X d,e≥1 (d,e)=t (h,dk_)=r (h,ek)=s 1 dk_ek η2k_(w) t2k_ηk_(r)ηk_(s), which yields (1.4.12) G(h) X (k) w|h X (k) r,s|h (r,s)=w X t≥1 (tk_,h)=w rsη2k(w) tk_wηk_(r)ηk_(s) h X a,b=1 tk_|a−b r|a+β1 s|b+β1 1 − h X a,b=1 tk_|a−b r|a+β2 s|b+β2 1 .

To estimate the absolute value appearing in (1.4.12), we will be using three distinct arguments. Firstly, we observe that (tk_{, r) = (t}k_{, s) = w} and thus, after a change of

variable, that (tk_ηk_{(w)/w, r) = (t}k_ηk_{(w)/w, s) = 1}. So it is enough to estimate the

error term of the double sum

Σ := A+N X a=A+1 B+M X b=B+1 l|ua+vb 1

with integers A, B, M, N, l, u, v ≥ 1 such that (u, l) = (v, l) = 1. For this, we write Σ = 1 l l X j=1 A+N X a=A+1 B+M X b=B+1 el(j(ua + vb)) = M N l + E and then E := 1 l l−1 X j=1 A+N X a=A+1 B+M X b=B+1 el(j(ua + vb)) l l−1 X j=1 1 kjukl 1 kjvkl l, where the last inequality follows from the Cauchy-Schwarz inequality.

Also, one can see that the solutions in our specic case, counted by one of the two expressions in absolute value in (1.4.12), represent points in a square of h by h points in

(27)

Z2 where the coordinates have the form (ir, js) and the condition tk|a − bis interpreted as the fact that these points are on lines of equation y = x + mtk_{. In view of this}

representation, we see that the main diagonal (m = 0) brings no contribution to G(h), and we deduce that the range of the sum on t in (1.4.12) is 1 ≤ t < h1/k_{. Also, for each}

t, the error is no larger than the number of non-main diagonals since the solutions are in arithmetic progression on each diagonal line. So the estimate h

tk is always true.

Using the above explanations, we see that the sum over t appearing in (1.4.12) is = h1/k X t≥1 (tk_,h)=w 1 tk h X a,b=1 tk_|a−b r|a+β1 s|b+β1 1 − h X a,b=1 tk_|a−b r|a+β2 s|b+β2 1 = 1 ηk_(w) h1/k_/η(w) X t≥1 (tk_ηk_(w)/w,h/w)=1 1 tk h X a,b=1 tkηk(w)|a−b r|a+β1 s|b+β1 1 − h X a,b=1 tkηk(w)|a−b r|a+β2 s|b+β2 1 1 w (hw)1/2k_/η(w) X t=1 1 + 1 η2k_(w) X t≥(hw)1/2k_/η(w) h t2k h 1/2k η(w)w1−1/2k.

Then, we easily verify that X (k) w|h w2k1 η2k−1(w) w2 X (k) r,s|h (r,s)=w rs ηk_(r)ηk_(s) Y p|h 1 + 1 p12 +1 p Y pj_kh (r−1)k<j≤rk 2r − 1 + 2p j prk ,

which shows in particular that G(h) h1/2k+_{as requested, thus completing the proof.}

Lemma 1.9. Let f be a function in Ck and consider the expression

S(q) := X e,d≥1 g(d) dk g(e) ek (dk_{, e}k₎2 (dk_{, e}k_{, q)}P M (dk_{, e}k_{, q)} (dk_{, e}k₎ (q ∈ N). It satises S(q) = 2ζ(1/k − 1) 1/k − 1 γf(q)M 1/k_{+ R(q),}

where γf(q) has been dened in the statement of Theorem 1.3 and where

(1.4.13) R(q) 2ω(q)Y p|q 1+1 p +M2k1 `₁(kθ_k) Y p|q 1+ 1 p1/2+ 1 p +MθkY p|q 1+ 1 pkθk .

(28)

Moreover, the last term in (1.4.13) can be omitted if kθk≤ 1/2.

Proof. We write t = (dk_{, e}k_{, q)}_{, d → dη(t), e → eη(t) and (d, e) = r}

S(q) = X (k) t|q 1 t X r≥1 (r,q/t)=1 r2k X e,d≥1 (d,e)=r g(η(t)d) dk g(η(t)e) ek P M t ηk_(t)rk

because if we write s = rη(t), then we have (dkη(t)k, ekη(t)k, q) = t ⇔ s k t , q t = 1 ⇔ s η(t) k ,q t = 1 ⇔ rk,q t = 1 ⇔ r,q t = 1. We thus have S(q) = X (k) t|q 1 t X r≥1 (r,q/t)=1 P M t ηk_(t)rk X e,d≥1 (d,e)=1 g(η(t)rd) dk g(η(t)re) ek = X (k) t|q 1 t X r≥1 (r,q/t)=1 ξ(η(t)r)P M t ηk_(t)rk = βf X (k) t|q 1 t X r≥1 (r,q/t)=1 θ(η(t)r)P M t ηk_(t)rk ,

From there, we use Lemma 1.6 as we did in Lemma 1.7. Firstly, regarding the main term, we have to evaluate

(1.4.14) X (k) t|q t1/k tη(t) X s|q/t µ(s) s X u≥1 ν(u) u (u, η(t)s)

(29)

in the range Mt ≥ [u, η(t)s]k_{. To do so, we complete the sum and consider as part of}

the error term the sum on Mt < [u, η(t)s]k_{. We easily obtain that}

X u≥1 ν(u) u (u, η(t)s) = Y p X i≥0 ν(pi) pi ! Y pα_kη(t)s X i≥0 θ(pα+i) pi ! X i≥0 θ(pi) pi !−1 ,

where we used the fact that X i≥0 ν(pi₎ pi = 1 −1 p X i≥0 θ(pi₎ pi 6= 0

for each prime p ≥ 2 and f ∈ Ck. Using this in (1.4.14), we obtain

Y p-q X i≥0 ν(pi₎ pi ! Y pα_kq (r−1)k<α≤rk 1 −1 p r−1 X i=0 θ(pi₎ pik + 1 pα−α/k+r X i≥0 θ(pr+i₎ pi !!

which yields for the main term of S(q) the expression 2ζ(1/k − 1) 1/k − 1 M 1/kY p-q 1 − 1 p X j≥0 1 pj |g(pj_)|2_{+ 2<g(p}j₎X i≥1 g(pj+i₎ pik × Y pα_kq (r−1)k<α≤rk 1 − 1 p r−1 X j=0 1 pjk |g(p j_)|2_{+ 2<¯}_g(pj₎X i≥1 g(pj+i) pik ! + 1 pα−α/k X i≥r 1 pi |g(p i )|2 + 2<¯g(pi)X l≥1 g(pi+l₎ plk !! .

It remains to show that the error term R(q) is (1.4.13). Firstly, for n = pα1

1 · · · p αi i , we compute X u>z |ν(nu)| u = X a1,...,ai≥0 |ν(npa1 1 · · · p ai i )| pa1 1 · · · p ai i X upa11 ···paii >z (u,n)=1 |ν(u)| u .

Using the upper bounds (1.4.7) we evaluate the above inner sum using the unique decomposition u = ab where µ(a)2 _{= 1} _{and b powerful, thereby allowing us to write}

X u≥w |ν(u)| u w X a=1 |ν(a)| a X b>w/a 1 b + X a>w |ν(a)| a 1 w1/2 w X a=1 |ν(a)| a1/2 + X a>w 4ω(a) ak+1 1 w1/2 + log3w wk 1 w1/2.

(30)

It follows that X u>z |ν(nu)| u 1 z1/2 X a1,...,ai≥0 |ν(npa1 1 · · · p ai i )| pa1/2 1 · · · p ai/2 i 1 z1/2 Y pkn |ν(p)| + |ν(p 2_)| p1/2 + |ν(p3_)| p + |ν(p4_)| p3/2 × Y pα_kn α≥2 |ν(pα_{)| +} |ν(pα+1)| p1/2 + |ν(pα+2_)| p 1 z1/2 Y pkn 1 p1/2 + 1 p + 1 p3/2 Y pα_kn α≥2 1 + 1 p1/2 + 1 p = 1 zγ1(n) 1/2 Y p|n 1 + 1 p1/2 + 1 p , where γ1(n) := Y pkn

p. In the same way, we have

X u≤z |ν(nu)| uξ            1 γ₁ξ(n) Y p|n 1 + 1 pξ if 1/2 < ξ < 1 l(ξ)z1/2−ξ γ1(n)1/2 Y p|n 1 + 1 p1/2 + 1 p if 0 < ξ ≤ 1/2 and X u≥1 |ν(nu)| u 1 γ1(n) Y pkn 1 + 5 p Y p2_|n 1 + 1 p . Gathering the estimates, for Mt < [u, η(t)s]k _{we have}

(31)

to the error term R(q). Then we verify that for Mt ≥ [u, η(t)s]k we have

thus completing the proof of Lemma 1.9.

Lemma 1.10. Let x ≥ 0 and h ≥ 1 be two integers and let S be a nite set of distinct numbers in [x + 1, x + h]. Further let I be a set of distinct positive integers. For each q ∈ I we choose a congruence class a mod q. Set

v(q) := #{u ∈ S| u ≡ a mod q} (q ∈ I) and

(32)

Further let W :=X q∈I v(q) =X u∈S r(u). Finally, let τI(n) := X q|n q∈I 1 for each n ∈ N, z := max u16=u2 ui∈S τI(u1− u2) and Xn := X (q1,...,qn)∈In qi6=qj f or i6=j 1 [q1, . . . , qn] .

Then, by noting S := #S and I := #I, we have the estimates (1.4.15) W ≤    2I + (2z)1/2_I1/2_S, n!S + n!1/nS1−1/n(hXn+ In)1/n (n ≥ 2).

Proof. For the rst inequality we write W =X q∈I v(q) ≤ I1/2 X q∈I v(q) +X q∈I v2(q) − v(q) 1/2 . Then we notice that if W ≥ 2I,

2IW ≤ W2 ≤ I X q∈I v2(q) , that is X q∈I v2(q) ≥ 2W = 2X q∈I v(q), which is X q∈I v2(q) − v(q) ≥X q∈I v(q). In this case we nd W =X q∈I v(q) ≤ I1/2 2X q∈I v2(q) − v(q) 1/2 . We see that v2_{(q) − v(q)}is the number of pairs (u

1, u2)with u1 6= u2for which q|u1− u2.

Then we have X q∈I v2(q) − v(q) = X u1,u2∈S u16=u2 τI(u1 − u2) ≤ zS2,

(33)

implying that

W ≤ 2I + I1/2(2zS2)1/2. For the second inequality we write

X u∈S r(u) ≤ X u∈S 1 1−1/n X u∈S r(u)n 1/n = S1−1/n X u∈S gn(r(u)) + X u∈S r(u)n− gn(r(u)) 1/n

where gn(w) :=Qn−1_j=0(w − j) is a polynomial of degree n, which we write as

gn(w) := n X k=0 s(n, k)wk. Clearly, s(n, n) = 1 and |gn(−1)| = Pn

k=0|s(n, k)| = n!. The s(n, k) are called Stirling

numbers of the rst kind. We use these for a combinatorial purpose: we want our n-tuples (q1, . . . , qn) to contain pairwise distinct numbers. We distinguish two cases:

(i) X u∈S gn(r(u)) ≥ K X u∈S r(u)n− gn(r(u)) (ii) X u∈S gn(r(u)) < K X u∈S r(u)n− gn(r(u))

for a positive constant K to be chosen later. In the rst case we deduce that X u∈S r(u)n=X u∈S gn(r(u)) + X u∈S r(u)n− gn(r(u)) ≤ 1 + 1 K X u∈S gn(r(u))

and we estimate the last sum by noticing that X u∈I gn(r(u)) = X (q1,...,qn)∈In qi6=qj f or i6=j

|{(u ∈ S| u ≡ ai (mod qi) for i = 1, . . . , n}|

≤ X (q1,...,qn)∈In qi6=qj f or i6=j h [qi, . . . , qn] + 1 = hXn+ In,

where we have used Lemma 1.3. In the second case we nd X u∈S r(u)n< (K + 1)X u∈S r(u)n− gn(r(u)) < (K + 1)(n! − 1) X u∈S r(u)n−1.

Now, we show by induction using the Cauchy-Schwarz inequality that for any j ∈ {0, . . . , n − 1}, X u∈S r(u)n< CX u∈S r(u)n−1 =⇒X u∈S r(u)j+1 < CX u∈S r(u)j,

(34)

where C > 0. We deduce that in this case we have W < (K + 1)(n! − 1)S. We have thus shown that

W ≤ (K + 1)(n! − 1)S + 1 + 1 K 1/n S1−1/n(hXn+ In)1/n

and the result follows with the choice K := 1 n!−1.

We say that an integer vector u = (u1, . . . , un) is primitive if (u1, . . . , un) = 1. The

following lemma can be seen as a generalization of Lemma 5 of [10].

Lemma 1.11. Let a1, . . . , am be a sequence of real numbers with each ai ∈ [A, 2A].

Assume that A, B, ∆ > 0 are real numbers satisfying

(1.4.16) ∆

A B.

Then the number of non zero integer solutions to the system of inequalities (1.4.17) |a1b1− aibi| ∆ (i = 2, . . . , m)

in primitive vectors b = (b1, . . . , bm) of length ≤ B is

(1.4.18) 1 + m−1 X j=1 B ∆ A j . Proof. We write vi := (a1, 0, . . . , 0, −ai, 0, . . . , 0) for i = 2, . . . , m.

The hypotheses imply that the vectors vi's span a vector space V of dimension m − 1.

Let v be a unitary vector that spans V⊥_{. The set {v, v}

2, . . . , vm} is a base of Rm. Let

b be any vector that satises (1.4.17). Setting

b := αv +

m

X

i=2

αivi,

we deduce from (1.4.17) that, for each i ∈ {2, . . . , m}, (1.4.19) hb, vii =

m

X

j=2

(35)

where θi is a real number no larger that 1 in absolute value and δi,j stands for the

Kronecker symbol. We deduce a linear system Mx = c where

M :=       a2 1+ a22 a21 · · · a21 a2 1 a21 + a23 · · · a21 ... ... ... ... a2 1 a21 · · · a21+ a2m       , x :=       α2 α3 ... αm       and c :=       θ2∆ θ3∆ ... θm∆       .

We show that the family of matrices dened by

An(x, z1, . . . , zn) :=       x + z1 x · · · x x x + z2 · · · x ... ... ... ... x x · · · x + zn      

satises the recurring relation (1.4.20)

det An(x, z1, . . . , zn) = z1det An−1(x, z2, . . . , zn) + z2det An−1(x, 0, z3, . . . , zn).

For this, we expand det An(x, z1, . . . , zn) with the rst row after having subtracted the

second row. By using the identity M−1_{det M = adj M}_{, (}_1.4.20_{) and a trivial estimate}

for the entries in adj M, we deduce that M−1 _{:= (c}

i,j) satisfy

|ci,j|

1

A2 for all 1 ≤ i, j ≤ m − 1.

We deduce from x = M−1_c_{that |α} i| ∆ A2 for 2 ≤ i ≤ m. Also, kb − αvk = h m X i=2 αivi, m X j=2 αjvji !1/2 = m X i,=2 αiαj(a21+ aiajδi,j) !1/2 ∆ A.

This proves that the solutions b to the system (1.4.17) are close to the line λv with λ ∈ R. More precisely, we write

K := {z ∈ Rn| kz − λvk ≤ C1∆

A and |λ| ≤ B}

for a large enough constant C1. Note that K is then a convex symmetric body. Let

(36)

solutions of the system (1.4.17) in K. If there is no solution then (1.4.18) holds. If j = 1 then there are at most two primitive solutions in K and the result follows. If 2 ≤ j ≤ m, then we write ¯K := K ∩ L. The integer points in L form a lattice Γ, in which case the number of solutions in ¯K is at most j·j!Vol( ¯_{det Γ}K) by a theorem of Blichfeldt (see [8] p. 62 (12)). Now, the result follows from the fact that det Γ ≥ 1 and from Vol( ¯K) B ∆_Aj−1, which in turn follows from inequality (1.4.16) and the fact that for any two orthogonal vectors u1, u2 in K, with ku1k ≥ ku2k, we must have

ku2k ∆_A.

Lemma 1.12. The estimate

Υ(h, r, s, m) := 2h X a,b=1 r|a,s|b m|a−b 1 h 2_{(r, s)} rsm + h(r, s) rs

holds when (r, s)|m and [r, s] h.

Proof. It is enough to consider, for a xed k, the equation

ar1− bs1 = km1 (r1 = r/(r, s), s1 = s/(r, s), m1 = m/(r, s)).

It is easy to show that two consecutive solutions a1 and a2 of this equation are distant

by exactly s1 and that |k| h/m. We deduce that

Υ(h, r, s, m) h rs1 + 1 h m + 1

and the result follows the hypothesis.

Lemma 1.13. Let f be a function in Ck. Then,

q X a=1 g(q, a, f )E(x; q, a, f ) 2 ω(q)_x1/k q (q ≥ 1). Proof. We have Σ := q X a=1 g(q, a, f )E(x; q, a, f ) =X s|q g(q, s, f ) q X a=1 (a,q)=s E(x; q, a, f ),

so that we are led to consider the sum X n≤x (n,q)=s f (n) = X n≤x/s (n,q/s)=1 f (ns)

(37)

= X (k) r|s X d≤zr,s (dk_,s)=r g(d) X n≤x/s (n,q/s)=1 dk_|ns 1 + X (k) r|s X d>zr,s (dk_,s)=r g(d) X n≤x/s (n,q/s)=1 dk_|ns 1 = Σ1+ Σ2,

say. First of all, Σ1 = X (k) r|s X d≤zr,s (dk_,s)=r (dk/r,q/s)=1 g(d) X m≤xr/sdk (m,q/s)=1 1 = X (k) r|s X d≤zr,s (dk_,s)=r (dk_/r,q/s)=1 g(d) φ(q/s)xr qdk + O(2 ω(q/s)₎ .

From there, we write φ(q/s)x q X (k) r|s r X d (dk,s)=r (dk/r,q/s)=1 g(d) dk + O x s X (k) r|s r X d>zr,s (dk_,s)=r (dk_/r,q/s)=1 1 dk + 2 ω(q/s)X (k) r|s zr,s η(r) = φ(q/s)g(q, s, f )x + O x s X (k) r|s r η(r)zk−1 r,s + 2ω(q/s)X (k) r|s zr,s η(r) . (1.4.21)

On the other hand,

holds for all f ∈ Ck, as can be seen directly from (1.1.10). Now, by using (1.4.21),

(1.4.22) and (1.4.23), we nd that |Σ| 1 q X s|q 2ω(q/s)τ (η(s))X (k) r|s zr,s η(r)

(38)

+x q X s|q τ (η(s)) s X (k) r|s r η(r)zk−1 r,s

so that with the choice zr,s=

xr s2ω(q/s) 1/k we obtain that |Σ| x 1/k q X s|q 2ω(q/s)(1−1/k)τ (η(s)) s1/k X (k) r|s r1/k η(r) x 1/k q X s|q 2ω(q/s)(1−1/k)τ 2_(η(s)) s1/k x 1/k q Y pj_kq 21−1/k 1 +X i≥1 (i + 1)2 pi/k x 1/k q Y pj_kq 21−1/k 1 + 4p 2/k (p1/k _{− 1)}3 2 ω(q)_x1/k q .

Lemma 1.14. Let q ≥ 2 be an integer and χ1 a multiplicative character modulo q. The

number of solutions χ to the equation χk _{= χ}

1, if any such solution exists, that is, if

χ1 can be written as a k-th power of a character, is given by the multiplicative function

g(k, q) := Y pj_kq g(k, pj) dened by g(k, pj) :=            (k, φ(pj)) if p ≥ 3, 1 if pj = 2, (k, 2) if pj _{= 4,} (k, 2)(k, φ(2j−1₎₎ _{if p = 2 and j ≥ 3.} In particular, g(k, q) ≤ 2kω(q). Lemma 1.15. Let q ≥ 2 be an integer. Then

X χ6=χ0 a+B X n=a+1 χ(n) 4 φ(q)B2_log2_q2ω(q)_.

Proof. This is the main result of the amazing paper of T. Cochrane and S. Shi [3]. The upper bound is obtained by a further smoothing in the case when B is small enough while careful computations are carried throughout.

(39)

1.5 Proof of Theorem

1.1

We refer to Lemma 1.1 with an:=Ph_a=1f (n + a) and T := cfh. We then have

(1.5.1) 1 N N −1 X n=0 h X a=1 f (n + a) = 1 N h−1 X n=1 nf (n) + h N X n=h f (n) + N +h−1 X n=N +1 (N + h − n)f (n) ! . Using the Abel summation, we nd that

h−1 X n=1 nf (n) = cfh2+ O(hEf(h)) − Z h 1 cft + Ef(t)dt = 1 2cfh 2_{+ O} hEf(h) + Z h 1 |Ef(t)|dt = 1 2cfh 2_{+ O(h}1+1/k_). Similarly, h−1 X n=1 (h − n)f (N + n) = 1 2cfh 2_{+ O(hN}1/k_).

Using these last two estimates along with (1.1.4) the sum in (1.5.1) can be written as 1 N N −1 X n=0 h X a=1 f (n + a) = 1 N cfhN + O(hN 1/k_{) = c} fh + O(hN1/k−1),

so that hN1/k−1 (by again referring to Lemma1.1). We deduce from (1.4.1) that

(1.5.2) M (h, N ) = N −1 X n=0 h X a,b=1 f (n + a)f (n + b) − |cf|2h2N + O(h2N1/k). We then have N −1 X n=0 h X a,b=1 f (n + a)f (n + b) = h X a,b=1 N −1 X n=0 f (n + a)f (n + b) (1.5.3) = h X a,b=1 N −1 X n=0 X dk_|n+a g(d) X ek_|n+b g(e) = h X a,b=1 X 1≤d,e≤(N +h)1/k g(d)g(e) N −1 X n=0 dk_|n+a ek_|n+b 1.

(40)

We focus our attention on the sum over d, e and we introduce a parameter z to split the sum in two parts. The rst is

Σ1(a, b) := X [d,e]≤z g(d)g(e) N −1 X n=0 dk_|n+a ek_|n+b 1

and the second is

Σ2(a, b) := X [d,e]>z g(d)g(e) N −1 X n=0 dk_|n+a ek_|n+b 1.

By using Lemma 1.3, we get that Σ1(a, b) = N X dk_,ek_≥1 (dk_,ek_)|a−b g(d)g(e) [dk_{, e}k_] + O X [d,e]≤z (dk_,ek_)|a−b 1 + N X [d,e]>z (dk_,ek_)|a−b 1 [dk_{, e}k_]

where the rst sum is what we have dened as γf(a, b) in (1.4.4). Now, by summing

over a and b, we obtain (1.5.4) h X a,b=1 Σ1(a, b) = N h X a,b=1 γf(a, b) + O h2z log N + hz log2N + h 2_{N log N} zk−1 + hN log2N zk−1 . Recall that the evaluation of the above sum was done in Lemma 1.7.

On the other hand, (1.5.5) h X a,b=1 Σ2(a, b) = h X a,b=1 X z<[d,e]<(N +h)1/k N −1 X n=0 dk_|n+a ek_|n+b 1 ≤ h#S, where S := {(m, n, d, e) : |mdk_{− ne}k_{| < h, md}k_{, ne}k _{< N + h, [d, e] > z}.}

We shall now describe a way to estimate S := #S. At rst, we count the solutions (m, n, d, e) of mdk = nek, mdk, nek < N + h, [d, e] > z. We see that it is (1.5.6) X [d,e]>z N [d, e]k N log2N zk−1 .

(41)

Then we count the remaining solutions that satisfy (m, n) = r and (d, e) = s

for some xed r and s. After a change of variables, we see that it corresponds to the set of solutions (m, n, d, e) of 0 < |mdk− nek_{| <} h rsk, (m, n) = 1, (d, e) = 1, de > z s, 1 ≤ md k_{, ne}k N rsk.

In particular, rsk _{< h}_{. So we x an X satisfying h X N and count the solutions}

with 0 < |mdk− nek_{| <} h rsk, (m, n) = 1, (d, e) = 1, de > z s, X rsk md k_{, ne}k X rsk.

We do a dyadic cut for each ot the variables d and e, so that we assume that d ∈ [L, 2L) and e ∈ [M, 2M) for some xed L and M that satisfy LM z

s. We apply the same

argument in two ways. First of all, we have 0 < m n − ek dk < h rsk_ndk hMk XLk.

For (d, e) xed, we have m

n in an interval of length hMk

XLk, and since (m, n) = 1, we

have a distance of at least M2k_r2_s2k

X2 between two solutions. We deduce that the

number of solutions is at most

(1.5.7) h Lk X Mk_r2_s2k + 1 = hX Lk_Mk_r2_s2k + 1. By summing (1.5.7) on (d, e) we obtain (1.5.8) hX Lk−1_Mk−1_r2_s2k + LM.

Also, there is no lost of generality in assuming that L ≥ M and then by using the mean value theorem to the function f1(w) = w1/k, we have

0 < m1/k n1/k − e d hM XL.

On the other hand, for xed (m, n), we nd that two distinct fractions e

d are separated

by at least 1

L2, so that the number of solutions is at most

(1.5.9) hM

XLL

2

+ 1 = hLM X + 1.

(42)

By summing (1.5.9) on (m, n) we obtain

(1.5.10) hX

Lk−1_Mk−1_r2_s2k +

X2 Lk_Mk_r2_s2k.

We sum over the domain LM z

s the common term in the two estimates (1.5.8) and

(1.5.10). This leads to

hX log N zk−1_r2_sk+1.

Summing over X and then over rsk _{< h}_{for a total of}

hN log N zk−1 .

For the two other terms in (1.5.8) and (1.5.10), we consider two cases depending whether LM is larger or smaller than _r2/(k+1)X2/(k+1)_s2k/(k+1). We have

X LM X2/(k+1) r2/(k+1)s2k/(k+1) X2 Lk_Mk_r2_s2k X2/(k+1)_{log N} r2/(k+1)_s2k/(k+1) and X z sLM X2/(k+1) r2/(k+1)s2k/(k+1) LM X 2/(k+1)_{log N} r2/(k+1)_s2k/(k+1).

We then sum over X and then over rsk_{< h} _{for a total contribution of}

h1−2/(k+1)N2/(k+1)log N

elements in the set S. It remains to consider the case where 1 ≤ mdk_{, ne}k h rsk. We

simply neglect the inequality and estimate with the number of elements (m, n, d, e) to nd X de>z/s h2 r2_s2k_dk_ek h2_{log N} r2_sk+1_zk−1

and, by summing over rsk _{< h}_{, we get to a total contribution of}

h2log N zk−1

which less than hN log N

zk−1 . So far, we showed that

(1.5.11) hS h 2_{N log N} zk−1 + hN log2N zk−1 + h 2−2/(k+1)_N2/(k+1)_{log N.}

It remains to put together (1.5.2), (1.5.4), Lemma 1.7 and (1.5.11) with the choice z = N1/k _{to complete the proof of Theorem} _1.1_.

(43)

1.6 Proof of Theorem

1.2

Proceeding as in the proof of Theorem1.1, we consider Lemma1.1with an:=Ph_a=1f (nh+

a + β) and T := cfh. We then have

1 M + 1 M X n=0 h X a=1 f (nh + a + β) = 1 M + 1 N +β X m=β+1 f (m) = 1 M + 1 cfN + O(N 1/k₎ = cfh + O(hN1/k−1),

from which it follows that (1.6.1) T (h, M; β) = h X a,b=1 M X n=0 f (nh + a + β)f (nh + b + β) − |cf|2hN + O(hN1/k).

say. We split Σ1 to get

(44)

We will also write

Ei := h

X

a,b=1

Ei(a, b) (i = 1, 2).

We verify that (t, η(r)) = (t, η(s)) = η(w), and also that [t, η(r)]|d and [t, η(s)]|e. Then, we use the upper bound in Lemma 1.12 and obtain

E1 X (k) w|h X (k) r,s|h X 1≤t≤zr,s,w (h,tk_)=w X [d,e]≤zr,s,w (dk_,h)=r (ek_,h)=s (d,e)=t h2 [r, s]tk + h [r, s] log NX (k) w|h X (k) r,s|h (r,s)=w X 1≤t≤zr,s,w (h,tk)=w h2_wη2_(w)z r,s,w rη(r)sη(s)tk+1 + hwη2_(w)z r,s,w rη(r)sη(s)t X (k) w|h X (k) r,s|h (r,s)=w h2_wz r,s,w rη(r)sη(s)η(w)k−1 log N + hwη(w)zr,s,w rη(r)sη(s) log 2_N. (1.6.2)

For E2 we distinguish the diagonal case a = b from the non-diagonal one. Hence we get

h X a,b=1 a6=b E2(a, b) X (k) w|h X (k) r,s|h X 1≤t≤h1/k (h,tk)=w X [d,e]>zr,s,w (dk_,h)=r (ek_,h)=s (d,e)=t h2 dk_ek log NX (k) w|h X (k) r,s|h (r,s)=w X 1≤t≤h1/k (h,tk_)=w η2_(w)h2 η(r)η(s)tk+1_zk−1 r,s,w

(45)

log NX (k) w|h X (k) r,s|h (r,s)=w h2 η(r)η(s)ηk−1_(w)zk−1 r,s,w and h X a=1 E2(a, a) X (k) w|h X (k) r,s|h X t≥1 (h,tk_)=w X [d,e]>zr,s,w (dk_,h)=r (ek_,h)=s (d,e)=t tkh dk_ek X (k) w|h X (k) r,s|h (r,s)=w X t≥1 (h,tk_)=w η2k_(w)h ηk_(r)ηk_(s)tkmin 1,η k−1_(r)ηk−1_(s)tk−1 η2k−2_(w)zk−1 r,s,w log N X (k) w|h X (k) r,s|h (r,s)=w X 1≤t≤zr,s,w η2(w)_η(r)η(s) (h,tk_)=w η2_(w)h η(r)η(s)tzk−1 r,s,w log N +X (k) w|h X (k) r,s|h (r,s)=w X t>zr,s,w η2(w)_η(r)η(s) (h,tk_)=w η2k(w)h ηk_(r)ηk_(s)tk X (k) w|h X (k) r,s|h (r,s)=w η(w)h η(r)η(s)zk−1 r,s,w log2N.

We now turn to the contribution of all the sums Σ2(a, b). We write

where S(r, s, t, w) is the number of 4-tuples (m, n, d, e) that satisfy |mdk_{− ne}k_{| < h, [d, e] > z}

r,s,w,

(46)

The evaluation of this quantity is done by essentially the same method that we have used for the evaluation of the quantity that appears in (1.5.5). First we need an upper bound for the number S0 of solutions to

mdk = nek, [d, e] > zr,s,w, (d, e) = t, (h, dk) = r, (h, ek) = s, 1 ≤ mdk, nek N. We nd that S0 X [d,e]>zr,s,w (d,e)=t (h,dk_)=r (h,ek_)=s N [d, e]k η2(w)N log N η(r)η(s)tzk−1 r,s,w .

Then, we count the number of solutions with (m, n) = j, noticing that jtk _{< h}_{. We}

denote this quantity by S(r, s, t, w, j). We thus x a parameter X which satises h X N. After a change of variables we have

0 < mη k_(r) ηk_(w)d k_{− n}ηk(s) ηk_(w)e k < h jtk, (d, e) = (m, n) = 1, X jtk m ηk_(r) ηk_(w)d k_{, n}ηk(s) ηk_(w)e k X jtk in the region de > η2_(w)z r,s,w

η(r)η(s)t . Now, we make a dyadic cut in each of the variables d and

e assuming that d ∈ [L, 2L) and e ∈ [M, 2M) in the region LM η2(w)zr,s,w

η(r)η(s)t . Again,

we use twice the same argument in two dierent ways. On the one hand, we have 0 < m n − ηk_(s)ek ηk_(r)dk hη k_(s)Mk Xηk_(r)Lk

and then, for (d, e) xed we have that two fractions m

n are distant by

j2_t2k_η2k_(s)M2k

X2_η2k_(w) .

The total number of solutions is therefore at most (1.6.3) hηk(s)Mk Xηk_(r)Lk X2_η2k_(w) j2_t2k_η2k_(s)M2k + 1 = hXη2k_(w) j2_t2k_ηk_(r)ηk_(s)Lk_Mk + 1.

Summing (1.6.3) on (d, e), we get to

(1.6.4) hXη2k(w)

j2_t2k_ηk_(r)ηk_(s)Lk−1_Mk−1 + LM

which is the rst wanted inequality. On the other hand, there is no lost in generality in assuming that Lη(r) ≥ Mη(s). Using the mean value theorem to the function g(w) = w1/k, we obtain 0 < m1/k n1/k − η(s)e η(r)d hM η(s) XLη(r) ⇒ 0 < m1/k_η(r) n1/k_η(s) − e d hM XL.

(47)

For (m, n) xed we have that two fractions e

d are distant by 1

L2 and thus the total

number of solutions is at most

(1.6.5) hM XLL 2_{+ 1 =} hLM X + 1. Summing (1.6.5) on (m, n) yields (1.6.6) hXη2k(w) j2_t2k_ηk_(r)ηk_(s)Lk−1_Mk−1 + X2η2k(w) j2_t2k_ηk_(r)ηk_(s)Lk_Mk

which is the second inequality.

We now consider the term that both estimates (1.6.4) and (1.6.6) have in common. Summing on LM η2_(w)z r,s,w η(r)η(s)t we nd hXη2_{(w) log N} j2_tk+1_η(r)η(s)zk−1 r,s,w . We then sum over X and then over j < h

tk to get

hN η2_{(w) log N}

tk+1_η(r)η(s)zk−1 r,s,w

.

For the two remaining terms in (1.6.4) and (1.6.6), we split according whether if LM is smaller or larger than

z := X

2/(k+1)_η2k/(k+1)

j2/(k+1)_t2k/(k+1)_ηk/(k+1)_(r)ηk/(k+1)_(s).

Proceeding in this way, we nd that X LM z X2η2k(w) j2_t2k_ηk_(r)ηk_(s)Lk_Mk X2/(k+1)η2k/(k+1)(w) log N j2/(k+1)_t2k/(k+1)_ηk/(k+1)_(r)ηk/(k+1)_(s) and X η2(w)zr,s,w η(r)η(s)t LM z LM X 2/(k+1)_η2k/(k+1)_{(w) log N} j2/(k+1)_t2k/(k+1)_ηk/(k+1)_(r)ηk/(k+1)_(s).

Summing over X and then over j ≤ h

tk we arrive at

hM2/(k+1)_η2k/(k+1)_{(w) log N}

tk_ηk/(k+1)_(r)ηk/(k+1)_(s) .

It remains to treat the case where 0 < mη k_(r) ηk_(w)d k_{− n}ηk(s) ηk_(w)e k < h jtk, (d, e) = (m, n) = 1, m ηk_(r) ηk_(w)d k_{, n}ηk(s) ηk_(w)e k h jtk

(48)

in the region de > η2_(w)z r,s,w

η(r)η(s)t . For that, we just ignore the inequality and we estimate

by the number of such elements (m, n, d, e). We nd X de>η2(w)zr,s,w_η(r)η(s)t h2η2k(w) j2_t2k_ηk_(r)ηk_(s)dk_ek h2η2(w) log N j2_tk+1_η(r)η(s)zk−1 r,s,w

and by summing over j < h

tk we get

h2_η2_{(w) log N}

tk+1_η(r)η(s)zk−1 r,s,w

which is less than hN η2_{(w) log N}

tk+1_η(r)η(s)zk−1

r,s,w. So far we have showed that

Σ2 X (k) w|h X (k) r,s|h (r,s)=w X 1≤t≤2N1/k (h,tk_)=w η2_{(w)N log N} η(r)η(s)tzk−1 r,s,w + hN η 2_{(w) log N} tk+1_η(r)η(s)zk−1 r,s,w +hM 2/(k+1)_η2k/(k+1)_{(w) log N} tk_ηk/(k+1)_(r)ηk/(k+1)_(s) = S1+ S2+ S3,

say. First of all,

S1+ S2 X (k) w|h X (k) r,s|h (r,s)=w η(w)N log2N η(r)η(s)zk−1 r,s,w + hN log N η(r)η(s)ηk−1_(w)zk−1 r,s,w

which amounts to the same contribution that we had obtained for the term with E2.

Then, S3 X (k) w|h X (k) r,s|h (r,s)=w hM2/(k+1)_{log N} η(k2_−k)/(k+1) (w)ηk/(k+1)_(r)ηk/(k+1)_(s) (1.6.7) hM2/(k+1)_{log N}Y p|h 1 + 2 pk/(k+1) . We now choose zr,s,w = M rs w 1/k . On the one hand we have

(1.6.8) S1 hM1/klog2N X (k) w|h X (k) r,s|h (r,s)=w w1−1/k_η(w) r1−1/k_η(r)s1−1/k_η(s) hM 1/k_log2_N

(49)

and on the other hand (1.6.9) S2 h2M1/klog N X (k) w|h X (k) r,s|h (r,s)=w w1−1/k r1−1/k_η(r)s1−1/k_η(s)ηk−1_(w) h 2_M1/k_{log N.}

We obtain the same estimate for (1.6.2) as for S1+ S2.

Gathering (1.6.7), (1.6.8) and (1.6.9) in (1.6.1), we obtain the estimate T (h, M ; β) = (M + 1) h X a,b=1 γf(h, a, b, β) − |cf|2hN +O h2M1/klog N + hM2/(k+1)log NY p|h 1 + 2 pk/(k+1) . (1.6.10)

Recalling (1.1.8) and Theorem 1.1, we obtain that (M + 1) h−1 X β0₌₀ h X a,b=1 γf(h, a, b, β0) − |cf|2h3− 2 ζ(1/k − 1) 1/k − 1 γfh 1+1/k ! = O N h1/2k+ N hθk_`(kθ k) + N2/(k+1)h2−2/(k+1)log N + h2N1/klog N +O h3M1/klog N + h2M2/(k+1)log NY p|h 1 + 2 pk/(k+1)

Letting M → ∞, we nd the estimate h−1 X β0₌₀ h X a,b=1 γf(h, a, b, β0) − |cf|2h3− 2 ζ(1/k − 1) 1/k − 1 γfh 1+1/k h1+1/2k_{+ h}1+θk_`(kθ k).

Finally, we use Lemma 1.8 at each term in the sum over β0 _{to get only terms with β}

and some error bounded by G(h). Then, we divide by h to get h X a,b=1 γf(h, a, b, β) − |cf|2h2− 2 ζ(1/k − 1) 1/k − 1 γfh 1/k h1/2k_{+ h}θk_`(kθ k) + G(h).

Theorem 1.2 follows by inserting this last inequality into (1.6.10).

1.7 Proof of Theorem

1.3

First observe that W(x; q, f ) =

q

X

a=1

(50)

= q X a=1 |Q(x; q, a, f )|2_{+ |g(q, a, f )x|}2_{− 2<Q(x; q, a, f )g(q, a, f )x} = q X a=1 |Q(x; q, a, f )|2_{− x}2_{|g(q, a, f )|}2_{− 2x<E(x; q, a, f )g(q, a, f )} = q X a=1 |Q(x; q, a, f )|2_{− x}2 q X a=1 |g(q, a, f )|2_{+ O} 2 ω(q)_x1+1/k q = W1− W2+ W3, (1.7.1)

say. The error term corresponding to W3 above follows from Lemma 1.13. Now, we

have W1 = q X a=1 x X m,n=1 m≡n≡a mod q f (m)f (n) = x X m,n=1 m≡n mod q f (m)f (n) = X |j|≤bx−1 q c x X n=1 1≤n+jq≤x f (n)f (n + jq). (1.7.2) For j = 0 we write x X n=1 |f (n)|2 _{= x} X d,e≥1 g(d)g(e) [dk_{, e}k_] + O X [d,e]≤x1/k 1 + x X [d,e]>x1/k 1 [dk_{, e}k_] = x X d,e≥1 g(d)g(e) [dk_{, e}k_] + O(x 1/k_log2_x). (1.7.3)

For each value of j 6= 0 we nd that

We further expand with

(1.7.4) Σ1(j) = (x − |j|q) X d,e≥1 (dk,ek)|jq g(d)g(e) [dk_{, e}k_] + O (E1(j) + E2(j)) ,

(51)

where E1(j) := X (k) w|q X (k) r,s|q X t≥1 (tk_,q)=w tk_|jq X [d,e]≤zr,s,w (dk_,q)=r (ek_,q)=s (d,e)=t 1 and E2(j) := x X (k) w|q X (k) r,s|q X t≥1 (tk_,q)=w tk_|jq X [d,e]>zr,s,w (dk_,q)=r (ek_,q)=s (d,e)=t 1 [dk_{, e}k_].

Using (1.7.4) in (1.7.2), we see that

(1.7.5) Σ := X |j|≤bx−1_q c (x − |j|q) X d,e≥1 (dk_,ek_)|jq g(d)g(e) [dk_{, e}k_]

will end up being the main contributing term to W1. On the other hand, a rst

A second one is given by X 0<|j|≤x/q E2(j) x X 0<|j|≤x/q X (k) w|q X (k) r,s|q X t≥1 (tk_,q)=w tk_|jq X [d,e]>zr,s,w (dk_,q)=r (ek_,q)=s (d,e)=t 1 [dk_{, e}k_] x log x X 0<|j|≤x/q X (k) w|q X (k) r,s|q (r,s)=w X t≥1 (tk_,q)=w tk|jq η2_(w) tη(r)η(s)zk−1 r,s,w

(52)

x 2_{log x} q X (k) w|q X (k) r,s|q (r,s)=w X t≥1 (tk_,q)=w t≤x1/k wη2_(w) tk+1_η(r)η(s)zk−1 r,s,w x 2_{log x} q X (k) w|q X (k) r,s|q (r,s)=w w η(w)k−1_η(r)η(s)zk−1 r,s,w . (1.7.7)

Now, we turn to the evaluation of the total contribution of the terms Σ2(j). We write

X 0<|j|≤x/q Σ2(j) ≤ X 0<|j|≤x/q X (k) w|q X (k) r,s|q (r,s)=w X t≥1 (tk_,q)=w X [d,e]>zr,s,w (dk,q)=r (ek,q)=s (d,e)=t x X n=1 1≤n+jq≤x dk|n,ek_|n+jq 1 (1.7.8) = X (k) w|q X (k) r,s|q (r,s)=w X t≥1 (tk_,q)=w S(r, s, t, w) where S(r, s, t, w) := #{(m, n, d, e)| 1 ≤ mdk, nek ≤ x, mdk _{≡ ne}k _{mod q,} [d, e] > zr,s,w, (d, e) = t, (tk, q) = w, (dk, q) = r, (ek, q) = s}.

For each 4-tuple (r, s, t, w) we may write (1.7.9) S(r, s, t, w) =X

u|q

T (r, s, t, w, u)

where T (r, s, t, w, u) counts the elements counted by S(r, s, t, w) restricted to the case where (mdk_{, q) = u}_{. After a change of variable, we notice that T (r, s, t, w, u) counts}

the points (m, n, d, e) that satisfy 1 ≤ mdk≤ xrη(w) k uηk_(r)tk, 1 ≤ ne k_≤ xsη(w)k uηk_(s)tk, de > zr,s,wη2(w) tη(r)η(s) ηk_(r)tk rη(w)kmd k_≡ ηk(s)tk sη(w)kne k _mod q u, ηk_(r)tk rη(w)kmd k_,q u = 1.

Also, we notice that [r, s] | u. A simple computation, not taking into account the congruence, shows that the estimate

T (r, s, t, w, u) x

2_rsη(w)2_{log x}

tk+1_u2_η(r)η(s)zk−1 r,s,w

(53)

is valid in each case, in particular when u = q. For the general case, we use Lemma

1.15 and obtain the estimate (1.7.10) T (r, s, t, w, u) g(k, q/u)x 2_rsη(w)2_{log x} φ(q/u)tk+1_u2_η(r)η(s)zk−1 r,s,w + g

1/2_{(k, q/u)2}ω(q/u)_x(rs)1/2_{η(w) log}3

x t(k+1)/2_uη1/2_(r)η1/2_(s)z(k−1)/2

r,s,w

. To prove it, we perform a dyadic cut for each variable d and e, assuming that d ∈ [L, 2L) and e ∈ [M, 2M) for some xed L and M such that LM ≥ zr,s,wη(w)2

tη(r)η(s) . In each such

region, we count the number of solutions with the characters modulo j = q/u, obtaining that 1 φ(j) X χ X d∼L,e∼M mx1/Lk nx2/Mk χ(αβ−1mdkn−1e−k) x1x2g(k, j) φ(j)Lk−1_Mk−1 + 1 φ(j) X χk_6=χ 0 X d∼L,e∼M mx1/Lk nx2/Mk χ(mdkn−1e−k)

where we have set x1 := xrη(w)k uηk_(r)tk, x2 := xsη(w)k uηk_(s)tk, α := ηk_(r)tk rη(w)k, β := ηk_(s)tk sη(w)k

and used Lemma 1.14. Now, by using the Cauchy-Schwarz's inequality three times (or the generalized Hölder's inequality) and Lemma1.15, we estimate the second term with

x1/2₁ x1/2₂ g1/2_{(k, j)}

L(k−1)/2_M(k−1)/2 log

2_q2ω(j)_.

It remains to sum this last inequality in the domain LM zr,s,wη(w)2

tη(r)η(s) to obtain the expected estimate (1.7.10). Now, observe that

X u|q [r,s]|u g(k, q/u) φ(q/u)u2 g(k, v) [r, s]2_φ(v) Y p|v 1 + 1 p and X u|q [r,s]|u 2ω(q/u)_{g(k, q/u)}1₂ u 2ω(v)_{g(k, v)}1₂ [r, s] Y p|v 1 + 1 p ,

where we have set v := q/[r, s]. We deduce from (1.7.9) that S(r, s, t, w) g(k, v)x 2_wη(w)2_{log x} qtk+1_η(r)η(s)zk−1 r,s,w Y p|v 1 + 2 p (1.7.11)

Topics in analytic number theory

Topics In Analytic Number Theory

Thèse

Patrick Letendre

Doctorat en mathématiques

Philosophiæ doctor (Ph. D.)

Québec, Canada

Topics in Analytic Number Theory

Résumé

Contents

Remerciements

Avant-propos

Introduction

Chapter 1

On a class of multiplicative functions

1.1 Introduction

1.2 Statement of the theorems

1.3 Some notation

1.4 Preparatory Lemmas

1.5 Proof of Theorem

1.1

1.6 Proof of Theorem

1.2

1.7 Proof of Theorem

1.3