ED 184 Mathématiques et informatique de MarseilleInstitut de Mathématiques de Marseille (UMR7373)

Mémoire présenté pour obtenirl’habilitation à diriger des recherches
Discipline : Mathématiques

Sary DRAPPEAU
Indépendance statistique et lois limites
pour quelques objets arithmétiques
Statistical independence and limit laws
for some arithmetical objects

Defended Septembre 28th 2023 in front of the jury composed of :Valérie BERTHÉ
Valentin BLOMER
Régis DE LA BRETÈCHE
Étienne FOUVRYFlorent JOUVE
Kaisa MATOMÄKI
Philippe MICHELJoël RIVAT
Emmanuel ROYERCNRSBonn Universität
Université Paris Cité
Université Paris Sud
Université de Bordeaux
Turun yliopisto
École Polytechnique Fédérale de Lausanne
Université d’Aix-Marseille
Université Clermont-Auvergne et CNRS

Numéro national de thèse/suffixe local : 2019AIXM0001/001ED62

Referee
Referee

RefereeThanks
Je suis tout d’abord redevable à Joël Rivat, Étienne Fouvry et Régis de la Bretèche pour leurs subtiles incitations, leurs précieux conseils et leurs encouragements durant tout le processus de HDR.
I’m grateful to Valérie Berthé, Valentin Blomer and Kaisa Matomäki for theirtime, and sincerely honoured that they agreed to report on the manuscript.
Je remercie vivement Florent Jouve, Philippe Michel et Emmanuel Royer pour
avoir accepté de participer au jury.
Je remercie encore Mme Tinel pour toute son aide au bureau des HDR.
Faire des mathématiques serait une activité bien morne si l’on ne pouvait lavivre avec les collègues. Je remercie donc toutes les personnes avec qui j’ai pu
partager l’envie de comprendre, l’enthousiasme et l’émerveillement, que cela ait
mené à un papier ou non. Je remercie en particulier Olivier Ramaré pour d’innombrables discussions éclairantes sur la théorie analytique des nombres ; etBrigitte Vallée pour avoir pris le temps de donner de nombreuses explications
sur ses travaux.
Je remercie enfin mes collègues de l’I2M pour l’atmosphère conviviale, qui
fait que ça vaut largement la peine de prendre le B1 en heure de pointe ou decohabiter avec des marteaux-piqueurs.

3

RésuméCe mémoire présente les thèmes sur lesquels ont porté mes travaux de recherche depuis mon arrivée à l’université d’Aix-Marseille en 2015. Leur problématique commune est de mettre en évidence des comportements statistiques
réguliers dans des familles d’objets arithmétiques naturels : les fonctions multiplicatives ou additives, et les valeurs centrales de certaines familles de fonctionsL.
La première partie concerne une question centrale en théorie multiplicative
des nombres : celle d’estimer la corrélation des valeurs f (n) et g(n + 1), où f et gsont deux fonctions multiplicatives, notamment lorsque l’une des deux fonctions
est la fonction « nombre de diviseurs ». Ce problème est naturellement lié à larépartition de certaines suites dans les progressions arithmétiques, et trouvent
des applications à d’autres questions arithmétiques, par exemple les zéros depetite hauteur des fonctions L de Dirichlet. Les majorations de sommes d’exponentielles algébriques sont un outil crucial dans cette partie du mémoire.
La seconde partie concerne certaines fonctions f : Q → C nommées par
Zagier « formes modulaires quantiques », caractérisées par certaines symétries
analogues à celles des formes modulaires. Mes collaborations sur ce sujet ont
consisté d’une part à établir ces relations de modularité quantiques dans certains
cas : celui de tordues additives de fonctions L de Dirichlet, et celui de sommes desymboles de Pochhammer ; et d’autre part à les utiliser pour en déduire, par desméthodes de systèmes dynamiques, l’existence de lois limites pour les valeursde f aux nombres rationnels ordonnés par dénominateurs croissants.
Mots clés : Fonctions multiplicatives, nombres premiers, progressions arithmétiques, sommes de Kloosterman, formes modulaires, formule de Kuznetsov,
tordue additive, forme modulaire quantique, loi limite, invariant de Kashaev.

4

Abstract
This manuscript presents the themes of my research works in Aix-Marseille
university since 2015. Their common theme is the search for simple statistical behaviour among families of natural arithmetical objects: multiplicative or
additive functions, functions defined in terms of numeration systems (decimal,
continued fractions...) and central values of L-functions.
The first part concerns a key question in multiplicative number theory: to estimate the correlation of values of f (n) and g(n + 1), where f and g are two multiplicative functions, with an emphasis on the case of the divisor function. Thisnaturally involves bounds on algebraic exponential sums, and leads to applications in various problems, all linked in some way to the distribution of certain
sequences in arithmetic progressions.
The second part concerns maps f : Q → C called by Zagier “quantum modular
forms”, which satisfy certain symetries analogous to those satisfied by modular
forms. In several collaborations, we established the modular quantum behaviour
in some cases related to additive twists of central L values, or to Pochhammer
symbols, and we deduced through methods from dynamical systems the existence of limit laws for values of f along rationals ordered by denominators.
Keywords: multiplicative functions, prime numbers, arithmetic progressions,
Kloosterman sums, modular form, Kuznetsov formula, additive twist, quantum
modular form, limit law, Kashaev invariant

5

Table des matières
Remerciements

3

Résumé

4

Abstract

5

Table des matières

6

Introduction

9

1 Exponential sums and statistical independence of arithmetic functions
1.1 Kloosterman sums . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Shifted convolution with the divisor function . . . . . . . . . . . . .
1.3 Zeroes of Dirichlet L-functions . . . . . . . . . . . . . . . . . . . . .
1.4 Friable values of quadratic polynomials . . . . . . . . . . . . . . . .

10
10
14
20
23

2 Modularity, continued fractions and limit laws
27
2.1 Statistical distribution of central values of additive twist L functions 28
2.2 Value distribution of a quantum knot invariant . . . . . . . . . . . 37
Bibliographie

41

6Introduction
This thesis presents the results I’ve obtained since 2015 in the course of myresearch in analytic number theory. Their commun setup is to find limit laws
among families of natural arithmetical objects : multiplicative or additive functions, functions defined in terms of numeration systems (decimal, continued fractions...), central values of families of L-functions.
The first part of the thesis is centered on the question of how to estimate
asymptotically the correlation sum
C(f, g; x) :=X

f (n)g(n + 1)n≤xwhen f and g are multiplicative functions. Heuristically, we expect the factorizations of n and n + 1 to be statistically independent on average over n. We
therefore expect C(f, g; x) to be comparable to the product of the averages of f
and g up to x, which means
C(f, g; x)  x−1 C(f, 1; x)C(1, g; x).
We will focus on the special case when f (n) = τ (n) = d|n 1, the divisor
function, and when the values g(p) have a simple description. This case is of particular importance because it is close to the threshold past which the analyticaltechniques currently known are ineffective. The most significant progress on thisquestion were made in the 1980’s, which the development of techniques fromautomorphic forms theory to study certain algebraic exponential sums which
naturally appear.
In Section 1.1, we present a generalization of results of Deshouillers and Iwaniec on the Fourier coefficients of modular forms, and bounds on averages ofKloosterman sums which follows by the Kuznetsov trace formula. These estimates are then used to deduce an arithmetical result on bounds on average overprimes of Kloosterman sums under the existence of Siegel zeroes, obtained with
J. Maynard.
In Section 1.2, we return to the estimation of C(τ, g; x), and we present works
which answer two natural questions : how the location of zeroes of Dirichlet Lfunctions influence the error term in the estimation of C(τ, g; x) ; and a simple
characterization of arithmetical functions g for which we are currently able toestimate precisely the correlation sum C(τ, g; x). The first work relies on theP

7bounds on Kloosterman sums mentioned above. The second result, obtened with
B. Topacogullari, depends on new combinatorial decompositions for certain multiplicative functions.
In Section 1.3, we present a work with K. Pratt and M. Radziwiłł in which
we study the density of zeroes of Dirichlet L-functions near the real axis, on
average over both the character and the modulus. The bounds for Kloostermansums refered to above will allow us to estimate this density for test functions
whose Fourier transform has a support larger than the range permitted by thelarge sieve inequalities.
Lastly, in Section 1.4, we present a work with R. de la Bretèche on the distribution of values of quadratic polynomials in arithmetic progressions, which alsorelies on the bounds for Kloosterman sums refered to above. As a corollary of thiswork, we improved a bit on a result of Deshouillers and Iwaniec on the largest
prime divisor of n2 + 1.
The second part of the thesis is centered on the existence of limit laws forcertain maps naturally defined on the rationals.
The first result on this second part concerns the distribution of values ofthe maps f : Q → C such that for any homography γ ∈ PSL2 (Z), the mapshγ (x) := f (x) − f (γx) are regular, in a certain precise sense. Such maps were called by Zagier “quantum modular forms” (of weight 0 and level 1) because someexamples came from quantum invariants in knot theory. By using the Euclid algorithm and dynamical properties of the Gauss map, we show under weak hypotheses on the maps hγ , that the values f (x) for x varying among rational numbersof denominators up to Q tend to distribute along a stable low as Q → ∞. This extends work of Baladi and Vallée regarding the hypotheses on hγ . This formalism
allows us to obtain a limiting law for rational values x sorted by denominatorsof the Estermann’s function, which is the value at s = 1/2 of the “additive twist”
of the function ζ(s)2 ,X τ (n)D(s, x) :=e2πinx .
sn≥1 n
This series does not have an Euler product ; however, the map x 7→ D(1/2, x) is aquantum modular form for which our results apply. The method involves mostly
arguments from dynamical systems, and the arithmetic is only invoked to prove
the quantum modularity of the map x 7→ D( 12 , x).
Our second result concerns the q-analogue of the Pochhammer symbol,(q)n =nY(1 − q r ),r=1

when q is a root of unity. We show that the maps which to x ∈ Q associate respectively (e(x))n and (e(x̄))n (where h/k = (h−1 mod k)/k (mod 1)) each satisfies,for γ ∈ P SL2 (Z), a form of modularity which relates their values at x and γx.

8These two formulas are close relatives of the modularity of the Dedekind η function. These formulas are consistent with the Zagier modularity conjecture on
Kashaev invariants of hyperbolic knots. We show, using these formulas, that the
Kashaev invariatns of the figure-eight knot
J41 ,0 (x) =X|(e(x))n |2n≥0have, for asymptotically almost all rational x (in a precise sense), an asymptotic behaviour which can be expressed in simple terms of the continued fraction
expansion of x.

9

1 Exponential sums and statisticalindependence of arithmetic
functions
1.1 Kloosterman sums
1.1.1 Bounds on average
For all q ∈ N>0 and a, b ∈ Z/qZ, define the Kloosterman sum (Kloosterman
1927) asX
an + bn−1
S(a, b; q) :=eqn∈(Z/qZ)×
where e(x) := e2πix , and n−1 denotes an inverse of n modulo q. We obviously
have the trivial bound|S(a, b; q)| ≤ ϕ(q).
Any improvement on this bound says something on the pseudorandomness ofthe sequence (n−1 ) as n varies in (Z/qZ)× . In full generality, we have the Weilbound (Weil 1948) : for any ε > 0,|S(a, b; q)| ε q 1/2+ε ,((a, b, q) = 1)

where (a, b, q) denotes the gcd. The exponent 1/2 is optimal. If a, b are non-zeroand fixed, we expect to be able to reach additionnal compensations in sums
over q of S(a, b, q). The first result of this type is due to Kuznetsov (Kuznetsov
1981), and asserts that for any ε > 0 and (a, b) ∈ Z2 \ 0,
X S(a, b; q)q≤xq

ε,a,b x1/6+ε .

This path has been developed in a systematic way by Deshouillers and Iwaniec(Deshouillers ; Iwaniec 1982a) (see also (Iwaniec 1982)). These ideas have prospered and contributed to some of the most precise results in modern analytic
number theory (Bombieri ; Friedlander ; Iwaniec 1986 ; Young 2011 ; Pitt 2013 ;Bettin ; Bui ; Li et al. 2020 ; Pratt ; Robles ; Zaharescu et al. 2020 ; Assing ; Blomer ;10

Li 2021 ; Maynard 2020).
The results of Deshouillers-Iwaniec (Deshouillers ; Iwaniec 1982a) are about
exponential sums of the following shape :
XXXXX
n

c

d

r



βn,r,s g(c, d) e ns(sc)−1,rd


where (βn,r,s ) is a finitely supported sequence, and g is a smooth compactly supported function. The case when βn,r,s = 1 if n = r = s = 1 and 0 otherwise,
which means that the sums runs in effect only over c and d, corresponds to the
Kloosterman sums case considered by Kuznetsov (Kuznetsov 1981), upon using
Poisson summation on the variable d. The quintuple sum above is a natural generalization to a realistic setup, where the “smooth” variables c and d are perturbed
by multiplicative convolution.
The following result generalizes Theorem 12 of (Deshouillers ; Iwaniec 1982a).
It was proved in (Drappeau 2017), and then made more precise in (Drappeau ;Pratt ; Radziwiłł 2022).
Théorème 1 (Drappeau 2017). Let C, D, N, R, S ≥ 1 and q, c0 , d0 ∈ N with (q, c0 d0 ) =1. Let (bn,r,s ) be a sequence supported in [1, N ] × [R, 2R] × [S, 2S] ∩ N3 . Let g :5∗ 3
R+→ C be a smooth function compactly supported in [C, 2C] × [D, 2D] × (R+)
which satisfies the bound∂ ν1 +ν2 +ν3 +ν4 +ν5 g(c, d, n, r, s) ν1 ,ν2 ,ν3 ,ν4 ,ν5 {c−ν1 d−ν2 n−ν3 r−ν4 s−ν5 }1−ε0 (1.1)∂cν1 ∂dν2 ∂nν3 ∂rν4 ∂sν5
for some ε0 > 0 and all νj ≥ 0. Then
XXXXX



bn,r,s g(c, d, n, r, s) e n

c
s
d n r
c≡c0 et d≡d0 mod q(qrd,sc)=1

¯
rd
sc

ε,ε0 (qCDN RS)ε+O(ε0 ) q 3/2 K(C, D, N, R, S)kbN,R,S k2 ,where kbN,R,S k22 =2n,r,s |bn,r,s | , andP

K(C, D, N, R, S)2 = qCS(RS + N )(C + RD)
qC 1−4θ
+ C 1+4θ DS((RS + N )R)1−2θ (1 + RD)
+ D2 N R.

(1.2)

Note that in a recent paper, Assing, Blomer and Li (Assing ; Blomer ; Li 2021)
prove a result which generalizes Theorem 12 of (Deshouillers ; Iwaniec 1982a)
in another direction, linked to the uniformity in n.
The main difference between theorem 1 and (Deshouillers ; Iwaniec 1982a,
Theorem 12) is the fact that the variables c, d, attached to a smooth weight, canbe restricted to arithmetical progressions. The main idea which makes this pos-11

sible was introduced by Blomer and Milićević (Blomer ; Miličević 2015) : it is to
exploit the possibility of varying the central character of the automorphic forms
for Γ0 (q) underlying the analysis in (Deshouillers ; Iwaniec 1982a). The successof the method relies on a suitable choice of Poincaré series, the Fourier expansionof which around well-chosen cusps will spawn the desired Kloosterman sums.
The usefullness of this setup was confirmed in many recent works on modular L
functions, where the Kuznetsov formula plays a rôle : let us cite (Kıral ; Young2019), (Petrow ; Young 2020) and (Zacharias 2019).
A difference which is less essential is the dependence in θ, which is made explicit, whereas the bound θ ≤ 1/4 from Selberg was used in (Deshouillers ; Iwaniec
1982a). This only affects the second term in (1.2). In most of the applications,this term is not the limiting one, but this is not systematically the case, as we will
see in theorem 6 below.
Due to bounds of the type of theorem 1, Kloosterman sums play a prominentrole in analytic number theory. It would be very desirable to have at our disposal bounds of the same quality for other exponential sums, such as Birch sums,
hyper-Kloosterman sums, or exponential sums whose phase parametrizes solutions to n3 ≡ 2 mod p. Recent works on such a setup was done by Buttcane,see (Buttcane 2022).

1.1.2 Average value along prime numbers and Siegel zeroes
The work of Blomer and Milićević (Blomer ; Miličević 2015), which we refered
to above, allows one to bound the sums
X

χ(q)S(a, b; q),

(1.3)

q≤x

in which χ is a Dirichlet character. It is natural to wonder if the extra flexibility allowed by the character χ could give additional arithmetic applications We
would be particularly happy to be able to prove the following statement,
X S(a, b; p)

√

p≤x

p

?

= o(x/ log x),(1.4)

where p denotes a prime number.
The question (1.4) is still wide open. It would follow from the “horizontal”
Satō-Tate conjecture, made by Katz (Katz 1980). In 2007, Fouvry and Michel (Fouvry ; Michel 2007) have obtained a non-trivial bound when the primality condition is relaxed to the condition that n belongs to the set Pk of integers having
at most k prime divisors, for some fixed k. Their method has been the subjectof many improvement, and the best known value of an admissible k is k = 7 by
Xi (Xi 2018).

12

The possibility to exploit the dependence in the character χ in (1.3) allows foran application, should there exist a Dirichlet character which takes the value −1
abnormally often on prime numbers. More precisely, we know that there is aconstant c > 0 such that for all Q ≥ 2 and each Dirichlet character χ primitive
modulo q ≤ Q, the inequality
L(1, χ) ≥ c/ log q
holds with the exception of at most one such character χ, which is then quadratic.
We conjecture that no such exceptions exist, and in fact the Riemann hypothesis
generalized to Dirichlet L functions implies the inequality L(1, χ)  1/ log log q ;but no substantial progress has been made on this question since it arose. In fullgenerality, the Siegel bound L(1, χ) ε q −ε for all ε > 0, which is non-effective
as soon as ε < 1/2, is the best-known one.
Suppose there is a sequence of primitive and quadratic Dirichlet characterχn (mod Dn ) such that L(1, χn ) log Dn → 0 as n → ∞. These characters would
then give a family of multiplicative functions which “mimick” the Möbius function on most integers on size DnO(1) without large prime divisors, while at thesame time, being of a controlled analytic complexity, since χn is periodic modulo Dn .
This idea was pursued in a seminal work of Heath-Brown in 1983 (HeathBrown 1983), and in many recent works of Friedlander and Iwaniec (Friedlander ; Iwaniec 2004 ; Friedlander ; Iwaniec 2005 ; Friedlander ; Iwaniec 2013). It
follows from these works that the existence of Siegel characters would allow tosolve many problems which are just beyond the threshold of modern analytic
techniques : the twin prime conjecture (Heath-Brown 1983), or primes in arithmetic progressions of large moduli (Friedlander ; Iwaniec 2003), or in very shortintervals (Friedlander ; Iwaniec 2004).
With James Maynard, we have proved in (Drappeau ; Maynard 2019) thatthe existence of exceptional characters would imply a non-trivial bound for the
average value of Kloosterman sums along prime numbers.
Théorème 2 (Drappeau ; Maynard 2019). For any ε > 0, there exist A, B > 0such that for any primitive quadratic character χ (mod D) and all x ≥ DA , wehave
X S(1, 1; p)≤ π(x) ε + BL(1, χ) log x).
√
p
p≤x
√
Since |S(1, 1; p)| ≤ 2 p by the Weil bound, the above statement is non-trivial
as soon as there exists a sequence χn (mod Dn ) of primitive quadratic characterswhich satisfies L(1, χn ) log Dn → 0, in other words, under the existence of Siegel characters. The estimate from theorem 2 is then relevant in a range of the
shape [DA , DC ] with C arbitrary but fixed.

13

The basic strategy is the one followed by Heath-Brown (Heath-Brown 1983)
and Friedlander-Iwaniec (Friedlander ; Iwaniec 2005 ; Friedlander ; Iwaniec 2004),
which consists in approaching the van Mangoldt function Λ(n) = log ∗µ(n) by
the convolution log ∗χ(n). From the point of view of analytic complexity, thislast function is comparable to 1 ∗ 1(n), the divisor function. The problem thentranslates to
X
X
S(1, 1, mn).
N <n≤2N M <m≤2M

When M < N 1−ε ou N < M 1−ε , the estimates from Deshouillers-Iwaniec, which
are of the same kind of theorem 1, give an acceptable bound. The main difficultyin the method is the case M = N . In this case, we are not able to exploit cancellation in the sign of the Kloosterman sums. The crucial√idea is to replace log(n/d)
Pin the convolution Λ(n) = d|n µ(d) log(n/d), by log( n/d). This has strictly noeffect on the rest of the arguments,
but has the advantage of dampening the√effect of those d of size about n, which correspond to the problematic caseabove. The rest of the argument follows a method of Fouvry-Michel (Fouvry ;Michel 2003), based on an idea of Hooley (Hooley 1964) and the numericalcoincidence 8 < 3π, which shows that |S(1, 1, n)| becomes negligible, on averageover n, as n has more and more prime divisors. This idea arises in many otherworks on Kloosterman sums, notably the recent work of Xi (Xi 2020) on the
non-coincidence between Kloosterman sums and Hecke-Maass eigenvalues.

1.2 Shifted convolution with the divisor function
1.2.1 The error term in the Titchmarsh divisor problem
Bounds on exponential sums of the same type as theorem 1 naturally findapplications to the estimation of “shifted convolutions”
X

f (n)g(n + 1),

(1.5)

n≤x

where f and g are assumed to be multiplicative, or to have a strong link with amultiplicative functions, such as the characteristic function of prime numbers.
One approach in this problem is to approximate f and g by non-trivial Dirichlet convolutions, which amounts to factorizing n = ab and n + 1 = cd, andthen to count solutions to the equation cd − ab = 1 weighted by some coefficients (Duke ; Friedlander ; Iwaniec 1994). In the best case scenario, the coefficients vary smoothly with a, b, c, d, and by Fourier analysis, we get back to
exponential sums of the same shape as Kloosterman sums. This explains why estimates based on Kloosterman sums are particularly interesting in multiplicative
number theory. We refer to (Kowalski 2003) for a closely related discussion.
The divisor function τ = 1 ∗ 1 can be seen as the most simple arithmetic

14

function which is not a perturbation of a constant function. The higher-orderdivisor functions τk = 1 ∗ · · · ∗ 1 (k times) are the most natural generalizations.
The indicator function of primes n 7→ 1P (n) can be seen as a sort of limit of
the maps τk , in the sense that if a question concerning a map f can be resolved
for f = τk , k arbitrary, with some uniformity in k, we expect that the samequestion could be resolved for f = 1P . This vague classification of multiplicative
functions materializes in “combinatorial identities” for multiplicative functions,the most well-known of which are those of Linnik, Vaughan and Heath-Brown,
and which will be the topic of this section.
The Titchmarsh divisor problem (Titchmarsh 1930) consists in estimating thesumX
T (x) :=
τ (p − 1).
p≤x

It is a special case of the question (1.5). The best estimation known isx
T (x) = c1 x + c2 li(x) + O(log x)A




(1.6)

due independently to Fouvry (Fouvry 1985) and Bombieri-Friedlander-Iwaniec(Bombieri ; Friedlander ; Iwaniec 1986). The shape of the error term follows fromthe use of the prime number theorem in arithmetic progressions, and the SiegelWalfisz theorem. It is a natural question to ask if a hypothesis such as Riemann’s
hypothesis on the ζ function would allow one to have a power-saving error termin x, as is the case in the prime number theorem. The following result, obtenedin (Drappeau 2017), shows that the answer is yes.
Théorème 3 (Drappeau 2017). Assume the Riemann hypothesis generalized to all
Dirichlet L functions. Then there exists δ > 0 such that
T (x) = c1 x + c2 li(x) + O(x1−δ )(x ≥ 2).

This result is actually the one which originally motivated theorem 1. An effective value of δ was obtained by Tang (Tang 2020). It is natural to expect that the
optimal error term should be O(x1/2+ε ), but the best value of δ permitted by themethod is quite far away from that.
The strategy is very close to the work of Fouvry (Fouvry 1985) and BombieriFriedlander-Iwaniec (Bombieri ; Friedlander ; Iwaniec 1986), but the constraint
to obtain a power-saving error term forbids us to use certain techniques they use,from sieve theory notably. By taking these constraints into account, it transpiresthat the exponential sum one has to estimate has the shapen(sc)−1 (cd)−1bn,r,s g(c, d) e+rdqc,d,n,r,s
X



15



(1.7)

where q ≥ 1 is a relatively small integer, of the order of xε . Previous works (Fouvry 1985 ; Bombieri ; Friedlander ; Iwaniec 1986) used several reductions to restrict to the case q = 1, which brought the above sum into the range which
Deshouillers-Iwaniec’s work (Deshouillers ; Iwaniec 1982a)
could handle. But√
these reductions are made at a cost of the order of x e−c log x at best. However, having theorem 1 at our disposal, we may work directly with the original sum (1.7)
by separating the variables c and d according to their congruence classes modulo q, keeping power-saving error terms in x throughout the argument. This
leads to the proof of theorem 3.

1.2.2 Titchmarsh’s problem for other multiplicative
functions
Consider the sum (1.5) with f = τ . We know how to obtain an asymptotic
estimate for g = 1P , as we have seen in the previous paragraph. However, until
2017, nothing seemed to be known for functions which are considered to be of
the same difficulty, or simpler, such as g = 1B , the indicator function of the set B
of integers which are sums of two squares.
More generally, we expect to be able to handle the case of the map g = τz
for z ∈ C in the Titchmarsh divisor problem, because it is considered to be of
the same “analytic difficulty” as the indicator function of primes : for instance,
the map s 7→ ζ(s)z possesses singularities at zeroes of the Riemann function ζ,
which are of logarithmic nature.
We resolved this problem in a work with Berke Topacogullari (Drappeau ; Topacogullari 2019). The “analytic difficulty” which we mentioned above materializes, in our result, through a hypothesis that the function g behaves as a periodic
function when restricted to prime numbers.
Théorème 4 (Drappeau ; Topacogullari 2019). Suppose that the multiplicative
function g : N → C satisfies the following hypotheses :— For some D ∈ N, whenever p, q are prime numbers with p ≡ q (mod D),then g(p) = g(q).
— For some A > 0, we have |g(n)| ≤ τ (n)A for all n ∈ N.
Then for all B > 0, we haveX
1<n≤xg(n)τ (n − 1) = 2

XX√χ primitive q≤ xcond(χ)|D cond(χ)|qX
1g(n)χ(n) + OD,A,B (x/(log x)B ).
ϕ(q) q2 ≤n≤x(n,q)=1(1.8)

The sum over n in the right-hand side is easy to estimate by using classical facts
on the zero-free region of Dirichlet L-functions. Fouvry and Tenenbaum (Fouvry ;Tenenbaum 2022) have recently obtained a result which contains theorem (1.8)

16and yields new applications on the shifted convolution problem (1.5) for additive
functions f, g.
Compared with the arguments underlying the Titchmarsh problem (1.6), thepart which we improve upon is the one commonly referred to as the “combinatorial identity”. It is an argument which has been studied extensively for prime
numbers (Vinogradov, Linnik, Gallagher, Vaughan, Heath-Brown) ; we refer tothe survey paper by Ramaré (Ramaré 2013) for a much more detailed discussion of the history of these techniques. The new ingredient in theorem 4 is acombinatorial identity for a multiplicative function g satisfying the hypothesesof theorem 4. In fact we obtain two such identities.
The first is an identity of Heath-Brown’s type (Heath-Brown 1982) for ταwith α ∈ Q. The possibility that such a formula exists was suggested by anotheridentity, due to Robert C. Vaughan, for τ1/2 which was kindly communicated to
us by Hugh Montgomery thanks to Olivier Ramaré. This identity for τ1/2 , which
we do not display here, will appear in volume II of Multiplicative Number Theory,
but we have not managed to use it in the Titchmarsh problem. After some trialand-error, we finally found a suitable generalization of Heath-Brown’s identity.
To continue with the simple special case of the function τ1/2 , it has the shapeXam (ζ(s)1/2 Mx (s) − 1)m ζ(s)1/2 = ζ(s)1/2 +m≥4Xb` ζ(s)` Mx (s)2`−1 ,

(1.9)

`≥1

where (am), (b` ) are complex numbers, ζ is the Riemann zeta function, andMx (s) =X

τ−1/2 (n)n−sn≤x1/4is an approximation to ζ(s)−1/2 . The Dirichlet series on the left-hand side of (1.9)
has no terms of index ≤ x, while the series on the right-hand side contains ζ 1/2
on the one hand, and non-negative integer powers of ζ and of the truncatedseries Mx g on the other hand. The coefficients am and b` are obtained by first finding a polynomial of the shape P (X) = 1 + XQ(X 2 ), Q ∈ C[X], which vanishes
5
at 1 at order 4, for instance P (X) = 1 − 35
X + 35X 3 − 21
X 5 + 16
X 7 . We then
16
16
16
1/2
2
formally substitute X = ζ M in the equality P (X) = 1 + XQ(X ), and then we
multiply ζ 1/2 .
It is much less obvious to see that this method generalizes well to the function τα for α = u/v rational, with a good control in terms of v on the size of thecoefficients b` . This last point is crucial. We prove that such is indeed the case ;the size of b` is then essentially controlled by the (archimedean) size of |u/v|.
This allows us, as a first step, to estimate the sumXτα (n)τ (n + 1)

n≤x

17

uniformly in α ∈ Q, |α|  1, denom(α)  (log x)A , with an error term ofsize O(x/(log x)A ).
To tackle the general case τz , z ∈ C, |z|  1, the second idea we use is simplya Lagrange interpolation on the polynomial map
Xz 7→

τz (n)τ (n + 1).

n≤x

The simplicity of this approach hides, of course, the issue to control the dependency in x in the interpolation step. We first estimate the contribution of termsof degree ≥ C log log x by OC (x/(log x)C+Oz (1) ) for C ≥ 7. 1 The order of magnitude log log x is related to the typical value of the number of prime divisors ofintegers of size up to x (a theorem of Hardy-Ramanujan). Then, a natural choiceof sampling points consisting of rational numbers of denominators O(log log x),
in the Lagrange interpolation step, leads to an error term of the shape[error term at the sampling points] × exp([degree]),(1.10)

which is to say (logxx)A × exp(C log log x), with A arbitrarily large. Finally, we
obtain for all z an estimate with error term


OA,C

x
x
+
,
C+O
(1)
z
(log x)(log x)A−Cand a suitable choice of A and C concludes the argument. The success of thisapproach crucially relies on the dependence on the degree in (1.10).
The second proof we have obtained is very similar to Vinogradov’s original
works (Vinogradov 1937). It does not rely on an interpolation step, and is therefore somewhat simpler to set up. The starting point is the Linnik identity (Linnik
1963, Chapitre VIII), which is to write!

ζ z = (1 + (ζ − 1))z =

X
j≥0

z
(ζ − 1)j .
jAs we expand the right-hand side as a Dirichlet series, the coefficient of an integer n involves indices j of size O(log n), which is way too large in practice. The
idea now is to factor the y-friable part of integers at the level of Dirichlet series,
which is to say, to writeζ = ζy My ,ζy (s) =Y(1 − p−s )−1 ,My (s) =Y(1 − p−s )−1 .

p>y

p≤y

We then chose y = xε for some small value of ε > 0 to be determined. By using
1. The exponent is not asymptotically optimal but it suffices for us that it tends to infinity
with C.

18

the strategy of Linnik to the part with My , we obtain
!

Myz =

X
j≥0

z
(My − 1)j ,jand if we wish to detect the coefficient of an integer n ≤ x, the sum over j can betruncated at j < 1/ε. To conclude, we bring back the various factors ζy , which has
no impact on the argument as long as ε is sufficiently small. In practice ε < 1/4
does the job. This is due to the fact that the indicator function of friable integers
benefits from good factorization properties in the sense of Dirichlet convolution.
In the case of the divisor function, we obtain for instance the decompositionτz (n) =

X
0≤`≤3

c`,z

X

τz−` (n1 )τ` (n2 ),(1.11)(n ≤ x),n=n1 n2p|n1 =⇒ p≤x1/4

where
c0,z = 1 − 11z + z 2 − 16 z 3 ,6c2,z = − 32 z + 2z 2 − 12 z 3 ,c1,z = 3z − 52 z 2 + 12 z 3 ,c3,z = 13 z − 12 z 2 + 16 z 3 .

Starting from the formula (1.11), it is relatively simple to follow the argumentsPof Bombieri-Fouvry-Friedlander-Iwaniec. We check that ` c`,z = 1 (which isPconsistent with (1.11) at n = 1) and that ` `c`,z = z (which is consistent
with (1.11) when n = p > x1/4 ).
The uniformity with respect to the complex variable z allows us to use the Cauchy formula to obtain asymptotic formula relative to integers having exactly kprime divisors.
Corollaire 1 (Drappeau ; Topacogullari 2019). For 1 ≤ k  log log x, we haveXn≤x
ω(n)=kτ (n − 1) =J
x X
Pj,k (log log x)(log log x)k
+
O
x
,
J,ε
log x j=1(log x)jk!(log)J−1+ε

where for all j ≥ 1, Pj,k is a polynomial of degree at most k − 1.
For k = 1, we recover the theorem of Bombieri-Fouvry-Friedlander-Iwaniec
with an error term of the same quality. Note that the number of integers n ≤ xsuch that ω(n) > C log log x is C x(log x)C−1−C log C . The condition on k incorollary 1 is therefore natural, since it corresponds to the domain in which theerror term, which arises from an application of the Siegel-Walfisz theorem, isrelevant.
The case z = 1/2 in (1.11) allowed us to answer the question which initiallymotivated us.

19

Corollaire 2 (Drappeau ; Topacogullari 2019). If B denotes the set of numbers
which can be written as sums of two squares, then for some sequence of numbers (βj )j≥0 and all J ≥ 0, we haveXτ (n − 1) = x(log x)1/2

n∈B∩[2,x]

J−1
X

βj
+ OJ (x(log x)1/2−J ).
j
(log
x)
j=0

These results can still be applied if n − 1 is replaced by n − h with h ∈ Z, h 6= 0
and |h| ≤ xδ for some δ > 0.
In this topic, one would like to be able to evaluate the left-hand side of (1.8)
even in a situation when the mean value of f is negligible with respect to any
negative power of log x. The error term given in theorem 4 is then no longer
negligible with respect to the main term. The case of friable integers was treatedin (Fouvry ; Tenenbaum 1990 ; Drappeau 2015), in particular thanks to work of
Harper (Harper 2012) for small values of y. It is natural to ask the same question
for the set of integers which have exactly k prime divisors when k/ log log x → ∞,that is to say, an analogue of corollary 1 for these large values of k. The characteristic function of these integers enjoys good factorization properties, and it isreasonable to expect that some of the techniques used in (Drappeau 2015) could
be adapted to treat this case. The problem of replacing the function τ in the lefthand side of (1.8) by a more general function seems, on the contrary, currently
out of reach. The simplest case after τ is that of the ternary divisor function τ3 ;P
the asymptotic estimation of n≤x τ3 (n)τ3 (n + 1) is a major open question in analytic number theory, and essentially no approach has been remotely effective up
to now.

1.3 Zeroes of Dirichlet L-functions
In this section, we assume the truth of the Riemann hypothesis generalized to
Dirichlet L-functions.
The problem we discuss here concerns the vertical distribution of zeroes ofDirichlet L-functions. Let χ be a primitive Dirichlet character. We are interested
in the multiset of zeroes of its associated L-function in the critical line,Zχ := {γ ∈ R, L( 12 + iγ, χ) = 0},
and then to the multisets
[

Zq :=

χ (mod q)
primitifZ≤Q :=

[
q≤Q

20

Zq .

Zχ ,

Using methods due essentially to Riemann and Weyl, it is known thatTqcard{γ ∈ Zχ ∩ [0, T ]} ∼
log
,
2π
2π




(T → ∞, T ≤ q o(1) ).

Using this estimate, it is natural to expect to find, at a distance  1 from the realaxis, of the order of log q elements of Zχ . Very little is known for a given fixedcharacter χ, as q → ∞. However, we have precise conjectures when we let χ vary
among all Dirichlet characters modulo q, that is to say, if we average over Zq .
Given a map φ : R → C and an integer q ≥ 1, we are interested in the quantity
W (φ, q) :=

X

X



φ

χ (mod q) γ∈Zχprimitifγ
log q .
2πDenote by ψ(q) the number of primitive characters modulo q.
Conjecture 1. Suppose that the Fourier transform φ̂ is of compact support. Then
we haveZ
W (φ, q) ∼ ψ(q) φ(ξ) dξ
(1.12)
as soon as the right-hand side tends to ∞.
This conjecture is a special case of a vast net of conjectures made by KatzSarnak (Katz ; Sarnak 1999), which give a hypothetical link with the distributionof eigenvalues of random matrices. These conjectures find their roots in an article of Montgomery (Montgomery 1973) on the pair correlation of zeroes of the
Riemann ζ function.
Conjecture 1 and its generalizations can be sometimes established under hypotheses relative to the size of the support of the Fourier transform of φ. The
following result, which is proved for instance in (Sica 1998), follows relatively
easily from the generalized Riemann hypothesis.
Théorème 5. Under the Riemann hypothesis generalized to Dirichlet L-function,
the asymptotic equivalenc (1.12) holds true as long as φ̂ is supported inside ]−2, 2[.
In joint work with Kyle Pratt and Maksym Radziwiłł (Drappeau ; Pratt ; Radziwiłł 2022), we extend the condition on the support of φ̂, at the cost of having an
additional average over the modulus q.
Théorème 6 (Drappeau ; Pratt ; Radziwiłł 2022). Under the Riemann hypothesis
generalized to Dirichlet L-functions, we haveXq≤Q

W (φ, q) ∼

X

Z

ψ(q)

φ(ξ) dξq≤Q

if φ̂ is supported in ]−2 − δ, 2 + δ[ with δ = 50/1093.

21

A similar statement holds without having to assume the Riemann hypothesis,
one simply needs to replace γ by −i(ρ − 1/2) in the definition of W (φ, q).
The main feature of theorem 6 is that we pass beyond the threshold 2 on
the size of the support of φ̂, which could not be deduced in a simple way fromthe Riemann hypothesis as in theorem 5. The situation is analogous to that ofcounting primes in arithmetic progressions of modulus q, where a trivial application of the generalized Riemann hypothesis leads to a condition q ≤ x1/2−ε
for individual moduli, which we can beat unconditionally in some cases (Fouvry
1985 ; Bombieri ; Friedlander ; Iwaniec 1986) by performing an additional average over q. This analogy is more than just formal, since the same tools are atplay in both cases.
The method begins with an application of the explicit formula of RiemannWeyl, which takes essentially the shapeX

W (φ, q) ≈

X

q≤Q

Z

ψ(q)

q≤Q

log p
φ(ξ) dξ +χ(p)φ̂
.
log q
q≤Q χ (mod q) p
X

X

X





primitif

By orthogonality of Dirichlet characters, this last sum can be evaluated if weknow how to estimate
log p
ψ(q)
φ̂
.
log q
q≤Q
p≡1 (mod q)
X

X





We are then reduced to counting primes in arithmetic progressions. The fundamental problem is the following : prove that there exist real numbers κ > 2
and η > 0 such that we have
X
q≤Q

X
p≤X
p≡1 (mod q)

1−

√
li(X)
 Q1−η X,ϕ(q)lorsque X = Qκ .

(1.13)

This question may seem paradoxical. Indeed, in the estimates of Bombieri-Vinogradov’s
type, the range X > Q2+ε is the easy one, and the whole problem is to study smallvalues of X, whereas here we wish to take X somewhat larger than Q2 . The mainthing at stake√is in fact the error term, which we wish to show is negligible withrespect to Q X, whereas to study prime numbers, the bound to be beaten israther of size X. Then, for X somewhat larger than Q2 , the error term we wish
to achieve is of size X 1−η , whereas the Bombieri-Vinogradov theorem gives us anupper-bound of size X/(log X)A .
The “proof of concept” of theorem 6 is given by the estimate of theorem 3.

22This last theorem proves that if the generalized Riemann hypothesis is true, then
X
q≤Q

li(X)
1−
 X 1−η
φ(q)


X
p≤X
p≡1 (mod q)(X = Q2 )for some η > 0, which is a bound of type (1.13) for κ = 2. All the work consists inshowing that one may take κ slightly larger while keeping an error term of thesame quality.
The fact that we can make do without the generalized Riemann hypothesis intheorem 6 is related to the fact that the characters χ which are involved in thedefinition of W (φ, q) are all primitive. We can therefore take away, in our wholeargument, the contribution coming from characters of conductor, say, at most xδ
for some small δ > 0. These characters are the only obstruction to getting a
power-saving error term in X in the absence of a Riemann hypothesis.

1.4 Friable values of quadratic polynomials
Another emblematic application of the estimates of Deshouillers and Iwaniec(Deshouillers ; Iwaniec 1982a) concerns the level of distribution of quadratic
polynomials. This is due to the fact that the roots of a quadratic polynomial
naturally give rise to an exponential sum of the same type as Kloosterman sums.
The problem is to estimate the sum∆(x, Q, λ) =

X
q≤Q





X

λq

1 − TP(x, q) ,n≤xn2 ≡−1 (mod q)

where (λq ) is a sequence of complex numbers of moduli at most 1, and TP(x, q)
is the expected main term, which means
TP(x, q) :=

xρ(q)
,
q

ρ(q) = card{m (mod q), m2 ≡ −1 (mod q)}.

We may conjecture that for all ε > 0, there exists δ > 0 such that∆(x, Q, λ) = O(x1−δ ),(Q ≤ x2−ε ),uniformly in λ. In practice, we have q≤Q TP(x, q)  x log(2Q), and any boundon ∆(x, q, λ) which is negligible with respect to x log Q is interesting. This problem is more and more difficult as the value of Q gets larger and larger.
The following result was obtained in joint work with Régis de la Bretèche (LaBretèche ; Drappeau 2020).
P

23

Théorème 7 (La Bretèche ; Drappeau 2020). Assume that the sequence (λq ) is
well-factorable in the sense of Iwaniec. For all ε > 0, there exists δ > 0 such thatfor Q ≤ x1+25/178−ε .

|∆(x, Q, λ)|  x1−δThis improves quantitatively a result obtained by Iwaniec in 1978 (Iwaniec
1978), which dates from before “kloostermania”. Such a bound has consequenceson problems involving the factorization of integers of the shape n2 + 1 ; for instance, Iwaniec uses his estimate on |∆(x, Q, λ)| to prove, at the outcome of significant work, that the polynomial n2 + 1 takes, infinitely often, values with at mosttwo prime factors. This is an approximation of the Landau conjecture that n2 + 1
is prime for infinitely many values of n.
We applied theorem 7 to another problem, which goes back to work of Chebyshev. The question is to find values of n such that n2 + 1 has a large prime divisor.
The best result before 2017 was due to Deshouillers and Iwaniec (Deshouillers ;Iwaniec 1982b), who prove the lower bound
P+



Y



(n2 + 1)  x1.2024 ,x≤n<2xwhere P + (m) denotes the largest prime divisor of m, with the convention P + (1) =1. In the setting of the works which led us to theorem 7, we were able to update
what was known in this topic, in the light of the progress made on the Selberg
conjecture on small eigenvalues of the hyperbolic Laplacian on congruence surfaces. The explicit dependence on θ in the estimate (1.2), in particular, allows us
to deduce the following result.
Théorème 8 (La Bretèche ; Drappeau 2020). We have
P+



Y



(n2 + 1)  x1.2182 .

x≤n<2xThis result was then substantially improved by Merikoski (Merikoski 2022) :we now know that
 Y

+
2
P
(n + 1)  x1.279 .
x≤n<2xOne of the new ideas in the work of Merikoski is the use of a lower boundsieve of Harman in the arguments of Deshouillers-Iwaniec, which has the effectof substantially amplifying the efficiency of bounds on exponential sums in thisproblem.
Another part of our work with Régis de la Bretèche (La Bretèche ; Drappeau
2020), which in fact was the original motivation for this work, is the constructionof a lower bound sieve for the set of friable integers. An integer n is said tobe y-friable if P + (n) ≤ y. Friable integer are a recurrent theme in multiplicative24

number theory (Granville 2008 ; Hildebrand ; Tenenbaum 1993 ; Moree 2014),
and were a focus of my thesis work. DefineS(x, y) := {n ≤ x, p | n =⇒ p ≤ y},Ψ(x, y) := card S(x, y).
For all real u ≥ 1, it is known that
Ψ(x, x1/u ) ∼ xρ(u)(x → ∞),

where ρ : R≥1 → [0, 1] is the Dickman function. As u → ∞, we have asymptotically ρ(u) = u−(1+o(1))u . Analogously to prime numbers, we would like to be able
to count asymptotically the cardinality of those integers n ≤ x for which n2 + 1
is y-friable. This can be viewed as an instance of the problem (1.5) when f is thecharacteristic function of y-friable integers, and g is the characteristic function ofsquares ; or as a “dual” version of the conjecture of Landau which we mentioned
earlier. We conjecture that
C(x, x1/u ) := card{n ≤ x : n2 + 1 is x1/u -friable} ∼ xρ(2u),on the basis of the fact that n2 + 1 is typically of order of magnitude x2 , and on
the heuristic that such integers, which have no fixed divisors, take friable values
at a frequency similar to integers of the same size.
The question of obtaining an upper-bound on the quantity C(x, x1/u ) is simpler.
The best general upper-bound, due to Khmyrova (Hmyrova 1964), yields
C(x, x1/u )  xρ(u)for all ε > 0 and u  log x/ log log x, based on an elementary estimation of ∆(x, Q, λ)for Q ≤ x1−ε . In the paper (La Bretèche ; Drappeau 2020), we construct an upperbound sieve for friable integers, which gives, in conjuction with theorem 7, thefollowing estimate.
Théorème 9 (La Bretèche ; Drappeau 2020). For any fixed ε > 0, there exists c > 0such that for all 1 ≤ u ≤ c log x/ log log x, we have25
C(x, x1/u ) ε xρ((1 + 178− ε)u).

As u  log x/ log log x, we win asymptotically a power of x with respect toprevious estimates.
Theorem 7 gives two improvements with respect to the existing literature. The
first is about the numerical value of the exponent. The previous work in this topic
is due to Iwaniec, which obtained an analogous estimate with exponent 1/15, byusing the Weil bound on Kloosterman sums. Using the recent works on bounds
on Kloosterman sums on average, and notably theorem 1, is the main ingredient25

in the numerical gain. The second improvement in theorem 7 is much more
substantial but also slightly more technical : contrary to earlier works using the
Deshouillers-Iwaniec bounds, we have no need that the sequence (λq ) is supported on prime numbers or squarefree numbers. This flexibility comes from an
improvement in an argument of Duke-Friedlander-Iwaniec (Duke ; Friedlander ;Iwaniec 1995), and uses crucially the uniformity in q allowed in theorem 1.

26

2 Modularity, continued fractions
and limit laws
In this second part, the results we present concern objects originating fromarithmetic, which are studied using methods from dynamical systems.
We first describe the problem which initiated the works presented in this part.
It is about the seriesX τ (n)D(s, x) :=e(nx),sn≥1 n
where we recall that τ (n) is the number of divisors of n, and e(z) := e2πiz . This
series converges absolutely for Re(s) > 1 and x ∈ R arbitrary. When x ∈ Q,
Estermann (Estermann 1930) proved that the function D(·, x) has a meromorphic continuation to C, and a functional equation relating the value D(s, a/q)to D(1 − s, −d/q) when ad ≡ 1 (mod q). This can be seen, for instance, by expressing the periodic function n 7→ e(nx) in terms of Dirichlet characters.
The value D(1/2, a/q) is then closely linked with the central values L( 12 , χ)of Dirichlet L functions. Bettin proves in (Bettin 2016) that when q is a prime
number and (a, q) = 1, the twisted second moment
M2 (q, a) :=1

X

q 1/2 χ (mod q)

χ(a)|L( 12 , χ)|2

is close to the value at x = a/q of the central value of the Estermann function,
M2 (q, a) ' Re D( 12 , a/q) + Im D( 12 , a/q).

(2.1)

As Bettin remarks in (Bettin 2016), the estimation of M2 (q, a) is related to thatof the fourth moment of Dirichlet L functions,
M4 (q) :=X
1|L( 12 , χ)|4 ,ϕ(q) χ (mod q)

whose estimation with power-saving error term in q by Young (Young 2011), successively improved in (Blomer ; Fouvry ; Kowalski et al. 2017 ; Wu 2020), was aremarkable success of the methods described in the first part (algebraic exponen-27

tial sums, Kuznetsov formula). Indeed, we have by orthogonality of characters
M4 (q) =|M2 (q, a)|2 .

Xa (mod q)

In view of (2.1), this motivates the question of understanding the size of values
taken by D( 12 , x) as x varies in a set of rational numbers.

2.1 Statistical distribution of central values ofadditive twist L functions
2.1.1 Equidistribution of rational orbits for the Gauss map
Improving earlier results by Young (Kıral ; Young 2019) and Conrey (Conrey[s. d.]), Bettin (Bettin 2016) shows that the function
φ(x) := D( 12 , x) − D( 12 , − x1 ),(2.2)

initially defined for x ∈ Q \ {0}, extends to a Hölder continuous function on R \{0}. Along with 1-periodicity, this implies more generally that for any γ ∈ SL(2, Z),
the mapφγ (x) : x 7→ D( 12 , x) − D( 12 , γx)extends to a Hölder continuous function on R\{γ −1 ∞}. This shows that D( 12 , ·) isan instance of what Zagier called, in a seminal article (Zagier 2010), a “quantum”
modular form for SL(2, Z).
The equality (2.2) paves the way to an expression of D( 12 , x) in terms of thefinite orbit of the rational x under the Gauss map
T (x) = {1/x},

T : ]0, 1[→ [0, 1[,

where {·} denotes the fractional part of x, that is to say, using the Euclid algorithm :r(x)D( 12 , x) = D( 12 , 0) +X

φ((−1)j−1 T j−1 (x)),

(2.3)

j=1

where r(x) ≥ 0 is the smallest integer such that T r(x) (x) = 0. This equality brings
the problem entirely to that of the distribution on average of orbits of rational
numbers under the Gauss map. All the arithmetic in the problem is encoded in
the analytic properties of the map φ, which is usually called the “observable” inthe theory of dynamical systems.
Denote the set of rational numbers [0, 1[ of fixed denominator q by
Ωq := {a/q, 0 ≤ a < q, (a, q) = 1}.

28

The question we would like to be able to address is the following : if x is arational number taken uniformly at random from Ωq , what can we say of therandom variable D( 12 , x) as q → ∞ ?
The methods we have at our disposal do not give us any precise results on Ωq 1 .
The problem becomes tractable as soon as we perform an additional averageover q, that is to say, if we work in the larger set[Ω≤Q :=Ωqq≤Qof rationals of denominators at most Q. The distribution on average of Birkhoffsums of the shape (2.3), among rationals in Ω≤Q is a subject which recently
entered within the full scope of the theory of measured dynamical system, due
to the work of Vallée (Vallée 2000) and Baladi and Vallée (Baladi ; Vallée 2005).
Letting Ω≤Q be endowed with the uniform counting measure, we denote PQ
and EQ the associated probability and variance. One of the main results of Baladi
and Vallée in (Baladi ; Vallée 2005) is the following.
Théorème 10. Let (c(n))n≥1 be a sequence of real numbers satisfying c(n) =O(log n) for n ≥ 2. For any x ∈ Q ∩ (0, 1), we letr(x)f (x) =

Xc(aj (x)),

j=1where aj (x) = b1/T j−1 (x)c denotes the j-th coefficient in the continued fraction
expansion1x=.
a1 + a2 +1 1...

There exists δ > 0 and two functions U, V defined and holomorphic in a neighborhood W ⊂ C of 0, such that as Q → ∞, we have

EQ exp wf (x)







= exp U (w) log Q + V (w) + O(Q−δ )(w ∈ W ).

(2.4)

The quality of the error term in the right-hand side of (2.4) relies notably onan adapted version of a method of Dolgopyat (Dolgopyat 1998) which gives auniform spectral gap for the family of transfor operators associated with T ,∞X

1
1
Hτ : f 7→
f
2+iτ
n+x
n=1 (n + x)(τ ∈ R).

As explained in Section 1.1 of (Baladi ; Vallée 2005), analogues of theorem 10
1. We refer nonetheless to (Aistleitner ; Borda ; Hauke 2022) for some quite general recent
results.

29

were known and classical in the situation where x ∈ [0, 1] is chosen uniformly
at random according to the Lebesgue measure, and when the function studiedP
is fN (x) = N
j=1 c(aj (x)) with N → ∞. The parameter N plays formally the roleof log Q, up to a constant, which is the typical length of the continued fraction
expansion of a fraction in Ω≤Q . The analogous continuous problem is simpler bymany aspects : for instance, the continuous analogue of the estimate (2.4) can beobtained using information on perturbations of the single operator Hτ for τ = 0
only, which can be gotten using the theory of perturbation of operators, without
having to use the techniques of Dolgopyat.
In theorem 10, the function f (x) can be seen as a Birkhoff sum of the same nature as (2.3), where the observable is φ(x) := c(b1/xc). For the map D( 12 , ·) which
we are interested in here, the main difference with respect to the statement oftheorem 10 are the following 2 :1. our observable φ is Hölder continuous, rather than constant, separately on
1each interval [ n+1, n1 ],
2. our observable φ does not have an exponential moment, more precisely, we
have φ(x) ∼ cx−1/2 log(1/x) as x → 0.
In (Bettin ; Drappeau 2022b), we obtain a generalization of theorem 10 which
resolves both points. Given a map φ : [0, 1] → R, denoter(x)Sφ (x) =Xφ(T j−1 (x)),(x ∈ Q ∩ (0, 1)).

j=1

In the case of theorem 10, we therefore have f (x) = Sφ (x) for φ(x) = c(b1/xc).
In the case of the Estermann function, we have D( 12 , x) = D( 12 , 0)+Sφ (x) where φis a certain Hölder continuous map.
The statement of our result involves the integralIφ (t) :=Z 1(eitφ(x) −1)0dx(1 + x) log 2(2.5)(t ∈ R).

Théorème 11 (Bettin ; Drappeau 2022b). Let κ0 , λ0 , α0 > 0. Assume that thefunction φ : (0, 1) → R satisfies the following properties :— For each n ∈ N, the map φ|]1/(n+1),1/n[ extends to a κ0 -Hölder continuous
map,— We have
X 1 
n≥1 n

2

sup
1
1
x∈[ n+1,n]α0|φ(x)||φ(x) − φ(y)|λ0
+ sup
1
1
|x − y|λ0 κ0x,y∈[ n+1
,n
]



< ∞.

2. We neglect in what follows the effect of the sign change (−1)j−1 in (2.3).

30

(2.6)

Then there exists t0 , δ > 0 and two maps U, V : [−t0 , t0 ] → C such that






EQ exp itSφ (x)= exp U (t) log Q + V (t) + O(Q−δ ) ,(t ∈ [−t0 , t0 ]), (2.7)

as well as
12 log 2
Iφ (t) + Oε (t2 + |t|2α0 −ε ),π2
V (t) = Oε (|t| + |t|α0 −ε ).
U (t) =

If moreover we have α0 > 1, then there exists C ∈ R such that
U (t) =

12 log 2
Iφ (t) + Ct2 + Oε (t3 + |t|1+α0 −ε ).
2πThe conclusion of theorem 11 corresponds to the case t = iw of theorem 10of Baladi-Vallée. An estimate for t complex in the neighborhood of 0 can beobtained, with an identical proof, under the natural condition that φ admits exponential moments, in the sense that
X 1

sup

2
1
1
n≥1 n x∈[ n+1 , n ]

exp σ0 |φ(x)| < ∞for some σ0 > 0. This is equivalent to asking that φ(x) = O(|log(2/x)|), similarly
as in the growth hypothesis on φ in theorem 10. This hypothesis is not verified
in the case of the Estermann function.
The proof that we present with Bettin of theorem 11 relies on a modificationof the arguments of Baladi and Vallée (Baladi ; Vallée 2005), and particularly
the part which uses the works of Dolgopyat (Dolgopyat 1998). To resolve thefirst objection 1., we work in a space of Hölder continuous functions, which
can be done using standard analytical methods. This modification turned out tobe very useful to allow arbitrary large values of λ0 > 0 in condition (2.6). To
resolve the second objection 2., we use recent results of Kloeckner (Kloeckner2019) on the spectrum of perturbed linear operators, which are valid without
having to assume analyticity of the perturbation. The price to pay to obtain weak
hypotheses on φ is to express our results in terms of the integral Iφ (t) without
further evaluating it.
In a companion paper (Bettin ; Drappeau 2022a), we evaluate the integral Iφ (t)in most practical situations. In the most favourable cases, φ is square integrable,
and we can estimate the integral (2.5) by inserting a Taylar expansion of order 2.
We obtain the following corollary.
Corollaire 3 (Bettin ; Drappeau 2022b). Assume the hypotheses and notationsof theorem 11. Assume moreover that α0 ≥ 2, and that there does not exist a
constant c ∈ R and a map f on [0, 1] such that φ = c + f − f ◦ T . Then there

31exists µ ∈ R and σ > 0 such that for all t ∈ R, as Q → ∞, we have


PQ

2

Z t
Sφ (x) − µ log Q
e−v /2 dv√√≤t =+ o(1).
σ log Q−∞2π
The expression of µ here isµ=12 Z 1
dx
φ(x)
.
π2 0
1+x

The quantity σ, on the other hand, does not seem to admit in general a simple
explicit expression in terms of φ.
Lastly, we illustrate theorem 11 by a limit theorem on the sum of coefficients aj (x)of the continued fraction expansion,r(x)Σ(x) :=Xaj (x).

j=1

We obviously have Σ(x) = Sφ (x) for the choice φ(x) = b1/xc. This map φ is
not integrable near 0. To state the limit law, we define G1 as the cumulative
distribution function of the stable law S1 ( π6 , 1, 0), which is to say,G1 (v) :=Z v−∞

g1 (x) dx,

g1 (x) =1 Z ∞ −itx − π6 |t|− 122 it log|t|
π
e
e
dt.
2π −∞

(2.8)

We also denote γ0 the Euler-Mascheroni constant.
Corollaire 4 (Bettin ; Drappeau 2022b). For each v ∈ R, we have


PQ

Σ(x)
log log Q − γ0−
≤ v → G1 (v)log Q
π 2 /12

as Q → ∞.
The law S1 ( π6 , 1, 0) does not a have a moment of order 1. This corresponds tothe fact that φ(x) = b1/xc is not integrable on [0, 1].
The continuous analogue of corollary 4, that is, when x is taken uniformly at
random from [0, 1] (the sum Σ(x) being truncated at j ≤ N for a parameter N ∈N which tends to infinity), was established by Heinrich (Heinrich 1987).

2.1.2 Application to the Estermann function
We can now apply theorem 11 to the map D( 12 , ·). To simplify the exposition,
we consider its real part, which has the advantage of being an even function, andwe let ψ = Re φ where φ is defined by (2.2). As Bettin showed in (Bettin 2016),

32

the map ψ essentially takes the shapeψ(x) = x−1/2 (c1 log x + c2 ) + E(x),where E is Hölder continuous on R of exponent α for all α < 1/2, and c1 , c2 areconstants. In (Bettin ; Drappeau 2022a), the integral Iψ (t) is estimated byIψ (t) = 1 + iµt − σ 2 t2 | log t|3 + o(t2 | log t|3 )as t → 0+ , where µ ∈ R is a constant and σ = 1/π. By estimating the expectationof D(1/2, x) on Ω≤Q , we show that µ = 0. We then deduce the following result,which answers our initial question.
Corollaire 5 (Bettin ; Drappeau 2022b). For σ = 1/π, we have


PQ

2

e−v /2 dvq√≤t =+ o(1)−∞2π
σ (log Q)(log log Q)3
Re D(x)Z t

as Q → ∞.
The appearance of the additional factor (log log Q)3 , which we did not anticipate, is intimately linked with the fact that the map ψ is not square integrable.
This follows in fine from the presence of a double pole at s = 1 for the Dirichlet
series ζ(s)2 , of which D(s, x) is the additive twist.
The simplicity with which corollary 5 is deduced from theorem 11 obfuscates
our many attempts to obtain corollary 5 by other methods, which have all failed.
As an example, the most natural strategy was to pass through an estimation of
moments, in order to take advantage of orthogonality of additive characters. This
computation was performed by Bettin in (Bettin 2019) with the conclusion thatthe estimation is no longer relevant starting from the second moment, due to adominant contribution coming from an asymptotically negligible set of rational
numbers. To undertake this strategy would have required a fine understandingof the rationals which cause this phenomenon, and an analytically tractable wayof taking away their contribution. On the contrary, the dynamical methods which
underly theorem 11 analyze this obstacle in a natural and transparent way.
About the link with moments of Dirichlet L-functions, which we discussed at
the beginning of this section, it is disappointing that the exact relation (2.1), between the twisted second moment of Dirichlet L-functions and the Estermann
function, only holds when q is prime ; when q is not prime, this relation is perturbed by an additional multiplicative convolution. This does not allow us to transfer immediately the above results, which are valid on average over q without
constraints. We can nonetheless make a plausible conjecture on the distributionof values of the twisted moments M2 (q, a).

33

Conjecture 2. The multiset

M2 (a, q), 0 < a < q, (a, q) = 1
1/2(log q) (log log q)3/2



tends to distribute, as q → ∞, according to a centered Gaussian law.

2.1.3 Application to central values of additive twists ofmodular forms L-functions
Theorem 11 found an application to another type of maps on Q, which arebuilt out of modular forms. Let f be a cuspidal holomorphic modular form of
weight k and level N , which we denote f ∈ Sk (N ). Let
f (z) =

Xan n(k−1)/2 e(nz),(Im(z) > 0)n≥1

be the Fourier expansion of f at infinity. We define the L-function
L(f, s) =

X an
s
n≥1 n

,

initially on Re(s) > 1. It is known since Hecke that this map extends to an entire
function of s, which has a functional equation of the shapeL(f, s) = γf (s)L(f¯, 1 − s)where γf is essentially a quotient of values of Euler’s Γ function. When f is a
Hecke eigenform, the map n 7→ an is multiplicative, and L(f, s) then possesses
an Euler product. Hecke’s method consists in expressing L(f, s) as an integral, or
period, of the form f :L(f, s) =k−1(2π)s Z i∞f (z)(Im z)s+ 2 −1 dz,iΓ(s) 0(s ∈ C),(2.9)

where the integration path is a vertical line.
Analogously to the construction of the Estermann function, we can form theadditive twist :X anL(f, s, x) :=
e(nx)
s
n≥1 n
which converges for Re(s) > 1 and x ∈ R. We have an expression analogous tothat of Hecke (2.9) :k−1(2π)s Z i∞
L(f, s, x) =f (z)(Im z)s+ 2 −1 dz,iΓ(s) x34

(Re(s) > 1),

the integration path being again a vertical line. As x ∈ Q, the decay of f near x
along the half-line {x + iy, y > 0} allows to analytically extend the integral
on the right-hand side, and thus also the map L(f, ·, x), to the whole complex plane. The resulting function again satisfies a functional equation of the
shape L(f, s, a/q) ↔ L(f, 1 − s, −d/q) when ad ≡ 1 (mod q). We can define thecentral value
Lf (x) := L(f, 1/2, x)(x ∈ Q).
When k = 2 we find, up to a multiplicative factor, a quantity known as themodular symbol
Lf (x) = cf hxif ,hxif :=

Z i∞

f (z) dz(k = 2).

x

Modular symbols are used, in particular, to compute explicitely modular forms (Cremona 1997).
Motivated by applications related to the rank of the group of rational pointsof an elliptic curve along a family of extensions, Mazur and Rubin formulated
a series of conjectures on the distribution of values of modular symbols hxif
on average over x (Mazur ; Rubin 2021). Their original conjectures were madefor x with given fixed denominator, that is, for x ∈ Ωq with the notations of
the previous section. These conjectures are still open and presumed to be very
difficult. The computation of the second moment of hxif lies very close to thethreshold of known techniques (Blomer ; Fouvry ; Kowalski et al. 2023).
Still when k = 2, Petridis and Risager (Petridis ; Risager 2018) showed thatthe question could be resolved by performing an additional average over denominators, which means picking x ∈ Ω≤Q with the notations of the previous
section. Their main theorem applies to any given holomorphic modular form of
weight 2 on a Fuchsian group of the first kind, and implies in particular, for aform f ∈ Sk (N ), the existence of a constant σf > 0 such that for all t ∈ R,


PQ

2

Z t
Lf (x)
e−v /2 dv
√
√
≤t →
,
σf log Q−∞
2π
(Q → ∞).

(2.10)The method of Petridis and Risager (Petridis ; Risager 2018) uses the computation of a moment generating function, and is capable of giving a statement oftype “quasi-powers” analogous to (2.4).
When k = 2, one of the known properties of the modular symbol translates
into the fact that the map
x 7→ Lf (x) − Lf (γx)
is constant for any fixed γ ∈ Γ0 (N ) (cf. (Mazur ; Rubin 2021)). This property is
crucial in the setup of (Petridis ; Risager 2018). This certainly shows that x 7→
Lf (x) is a quantum modular form in the sense of Zagier.

35Consider now the case when the weight k ∈ Z is larger than 2. For all γ ∈Γ0 (N ), let
φγ (x) : x 7→ Lf (x) − Lf (γx)(2.11)
When k > 2, the map x 7→ Lf (x) − Lf (γx) for γ ∈ Γ0 (N ) is no longer necessarily constant. However, as it is proven in (Bettin ; Drappeau 2022b ; Nordentoft
2021a), all of the maps φγ extend to functions on R which are (1 − ε)-Hölder
continuous and bounded in x ∈ R. When N = 1, that is when f is modular withrespect to SL(2, Z), corollary 3 allows us to deduce the following central limit
theorem on the values Lf (x) for x ∈ Ω≤Q , which is an analogue of the theoremof Petridis-Risager where the hypothesis “f ∈ S2 (N )” is replaced by “f ∈ Sk (1)”.
Théorème 12 (Bettin ; Drappeau 2022b). Assume that f ∈ Sk (1). There exists σf >0 such that the estimate (2.10) holds for all t ∈ R, as Q → ∞.
The new aspect of our work is the fact that we use no information on f besides
a weak hypothesis on the regularity of φγ . It is the strategy of the method, basedon the expression in terms of iterates of the Gauss map, which allows this.
Lee and Sun (Lee ; Sun 2019), independently of us and roughly at the sametime, had the same idea of using Baladi-Vallée’s strategy on this problem, andcould handle the case when f ∈ S2 (N ). A main novel ingredient in their workis the idea to use a natural extension of the Gauss map to handle congruencegroups.
At about the same time, Nordentoft (Nordentoft 2021b) obtained the estimateof theorem 12 for any holomorphic modular form f on a Fuchsian group of thefirst kind, which includes theorem 12 as a special case. Nordentoft’s method
is different form Petridis-Risager’s, Lee-Sun’s and ours, but is more related toPetridis-Risager’s, and goes through the computation of moments.
In a recent work with Nordentoft (Drappeau ; Nordentoft 2022), we obtained
a result on the regularity on the maps (2.11) in the general case of a Maaß form
on a Fuchsian subgroup of SL(2, R) of the first kind, which generalizes both thecases of holomorphic forms (Nordentoft’s theorem) which we just discussed, andalso the situation of the Estermann function an = τ (n), which corresponds to a
special case when the Maaß form is a certain Eisenstein series. This allows us toprove a central limit theorem analogous to theorem 12 when f is replaced by a
Maaß form for the particular group SL(2, Z).
In a work in progress with Bettin and Lee, we are trying to work out an analogue of theorem 11 for the action of a general Fuchsian group of the first kindon the set of its cusps. When this work will have converged, we hope to have alimit law similar to (2.10) valid for all twisted L-values of L-functions of rank 2,
in the sense of (Iwaniec ; Kowalski 2004, Chapitre V).

36

2.2 Value distribution of a quantum knot
invariant
In Zagier’s article (Zagier 2010) where he introduces quantum modular forms,
a singular example arises from Kashaev invariants in knot theory.. The exact
definition of this invariant in general involves a very delicate combinatorialconstruction starting from a knot diagram. In the simplest non-trivial case, thatof the “figure eight” knot denotes 41 , this invariant takes the shape of a function J : Q → R given by
J(x) =∞ YnX|1 − e(rx)|2 .

(2.12)

n=0 r=1

The sum is finite when x ∈ Q. Zagier noticed numerically that the map h definedbylog J(x) − log J(1/x) = h(x)(2.13)
seems to extend to R \ {0} as a somewhat regular function, for instance continuous almost everywhere, whereas the function J itself is very irregular. These
two maps are drawn on figures 3 and 4 of (Zagier 2010).
Conjecture 3 (Zagier). — The map h extends to a function on R \ {0} whichis continuous at irrationals, and left- and right-continuous at rationals.
— We have h(x) = Cx − 32 log x + O(1) for x ∈ ]0, 1], where C = Vol(41 )/2π ≈0, 323 and Vol(41 ) is the hyperbolic volume of the figure eight knot complement.
This map gives the opportunity of an interesting application of theorem 11.
Indeed, by definition of h and using the value J(0) = 1, we haver(x)log J(x) =Xh(T j−1 (x)).

(2.14)

j=1

The function h does not exactly satisfy the regularity hypotheses stated in theorem 11, but it is not very far from it. If conjecture 3 was true, we would deducethe convergence in law to a stable law of parameter 1 similar to (2.8), for thevalues taken by log J(x) at rationals ordered by denominators.
We therefore went looking with Sandro Bettin for a proof of Zagier’s conjecture. This led us to study more precisely the q-Pochhammer symbol,(q)N := (1 − q) . . . (1 − q N ),(N ∈ N),(2.15)

whose modulus squared is involved in the definition (2.12). This sort of productis also called, in other contexts, a Sudler product (Sudler 1964), and we will
come back to this later on.

37

The symbol (q)N can be viewed as a truncated version of the Dedekind ηfunction,Yη(z) =(1 − e(nz)),(Im(z) > 0),n≥1which is a modular form of weight 1/2.
One of the two main results in the work (Bettin ; Drappeau 2022c) with Sandro
Bettin is a form of modularity for the q-Pochhammer symbol. We denote den(x)the denominator of x ∈ Q. Stated in an informal way, our result can be expressed
as follows.
Théorème 13 (Bettin ; Drappeau 2022c). Let γ ∈ SL(2, Z). For 1 ≤ r < den(γx),
we havexe( γx)(e(γx))r = χ(γ) e( 24)(e(x))r0 ψγ (x, r),(2.16)
24for some 1 ≤ r0 < den(x), where χ is the central character (“nebentypus”) associatedto the Dedekind function (Iwaniec 1997, p. 45), and ψγ is a map satisfying certain
holomorphicity conditions.
The variable with respect to which ψγ extends as a holomorphic function is z ={r den(x)/ den(γx)}. Our starting point is the summation formula of Abel-Planaapplied to the sumNXlog(1 − e(nz)).

n=1

The Abel-Plana formula is a version of the classical Euler-Maclaurin formula,where the error term is expressed in an exact way as an integral of the summand, here log(1 − e(z)). Our argument exploits, implicitely, the fact that thefunction log(1 − e(z)) is close to a primitive of the function π cot(πz), which isthe summation kernel associated to integers (it has a simple pole of residue 1 at
every integer).
The fact that formula (2.16) is exact, as well as the holomorphicity property
we mentioned about ψr , are two crucial aspects when one tries to sum this formula over r. Here we strongly relied on work of Ohtsuki Ohtsuki 2016. This has
allowed us to obtain a proof of Zagier’s conjecture for certain special cases ofhyperbolic knots 3 . In the case of the figure eight knot, we deduce the following
property.
Théorème 14 (Bettin ; Drappeau 2022c). There exists a sequence (cn ) of real
numbers such that for all Q, N ≥ 1 and x ∈ ]0, 1], if den(x) ≤ Q, we haveh(x) =

N−1
X
C 3− log x +cn xn + OQ,N (xN ).
x2
n=0

3. The few knots we study are essentially those for which the volume conjecture of Kashaev (Kashaev 1997) is known ; conversely, Zagier’s conjectures implies the volume conjecture.

38

Now equipped with such a property, we would like to be able to deduce,
through the definition (2.13), a limiting law for the values log J(x). Unfortunately, the fact that the error term depends implicitely on the denominator of x
forbids us to undergo this strategy at this point. We found an indirect successful
path, however, by complementing theorem 13 with another reciprocity formula.
Write x = [0; a1 , . . . , ar ] the continued fraction expansion of x. Applying theGauss map consists in taking away from x its first coefficient,
T (x) = [0; a2 , . . . , ar ].
In certain circumstances (Bettin ; Conrey 2013), it is analytically beneficial to try
to take away its last coefficient instead,
U (x) := [0; a1 , . . . , ar−1 ].
This amounts, up to a sign, to conjugating the Gauss map by the involution whichassociates to a fraction a/q ∈ ]0, 1[ the fraction (−1)r−1 ā/q (mod 1), where aā ≡1 (mod q). Indeed this involution has the effect of reversing the continued fraction expansion.
The Bezout relation
ā q̄
1
+ =
(mod 1)q a
aq
is a precise version of the assertion that the numbers x and U (x) are close toeach other, which can also be seen from the definition of U .
We obtain the following relation.
Théorème 15 (Bettin ; Drappeau 2022c). We havelog J(x) = log J(U (x)) + Car (x) + E(x),where E(x) is an error term bounded by log ar (x) in most circumstances.
This theorem is specific to the knot 41 , contrarily to theorems 13 and 14. Treating the error term required a specific technical lemma about partial sums of thecotangent function (Bettin ; Drappeau 2020), which was one of the most troublesome part of the argument.
The conjunction of theorems 13 and 15 yields just enough information on thefunction J to obtain, through the iteration (2.14), the following approximation.
Pr(x)We recall that Σ(x) = j=1 aj (x).
Théorème 16 (Bettin ; Drappeau 2022c). For all x ∈ Q ∩ ]0, 1[, we havelog J(x) ∼ CΣ(x),

39

Σ(x)
→ ∞.
r(x)

On the one hand, it is classically known that r(x)  log Q when den(x) ≤ Q.
On the other hand, corollary 4 ensures that
Σ(x) ∼

12(log Q) log log Q
π2

for a proportion 1 + o(1) of rationals x ∈ ΩQ 4 . We immediately deduce thefollowing corollary, which can be seen as a law of large numbers for the valuesof log J(x) for x ∈ ΩQ .
Corollaire 6 (Bettin ; Drappeau 2022c). There exists a map ε : N → R+with ε(Q) → 0 as Q → ∞, such that12log J(x) =C + O(ε(Q)) (log Q) log log Q
π2for all x ∈ ΩQ , with at most ε(Q)Q2 exceptions.
Very recently, Aistleitner and Borda obtained several results on J(x) using
their works on perturbations of products of the sine function (Borda ; Aistleitner 2022c ; Borda ; Aistleitner 2022b ; Borda ; Aistleitner 2022a). In particular,
it is proved in corollary 2 of (Borda ; Aistleitner 2022a) that the map h defined in (2.13) can be extended by density to a function which is almost everywhere continuous on R \ {0}. More precisely, Aistleitner and Borda show that
this extension of the function h is continuous at those irrationals x which are“well-approximable” in the sense that the sequence (aj (x))j is unbounded. This
allowed them to obtain, in theorem 4 of (Borda ; Aistleitner 2022a), a central
limit theorem for log J(x), of the same precision as corollary 4.
One of the main questions left open is the continuity of h at badly approximable rationals, in particular at quadratic irrationals such as 1/φ = [0; 1, 1, . . . ].
This is the topic of a work in progress with Sandro Bettin and Bence Borda. We
have good hope in the case of quadratic irrationals.
Another lead we wish to explore, but on which we have no serious angle of attack, is to the validity of the law of large numbers of the same type as corollary 6,
for more general knots. Our proof for the knot 41 uses crucially the positivity ofthe summands in (2.12), and a new idea is required in order to dispense with thisingredient. It is more precisely our proof of theorem 15 which does not seem tobe well-suited to other cases. The works of Aistleitner-Borda (Borda ; Aistleitner2022a), as well as our work in progress with Sandro Bettin and Bence Borda on
the function h, may give another starting point.

4. Actually this is true for a proportion 1 + o(1) of rationals x ∈ Ωq , without averaging overthe denominators, thanks to works of Rukavishnikova (Rukavishnikova 2011) ; see also the recent
paper (Aistleitner ; Borda ; Hauke 2022).

40

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