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# Granville, Andrew

Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-6583

### Affiliations

• Membre Centre de recherches mathématiques
• Titulaire Chaire de recherche du Canada en théorie des nombres
• Membre CRM  Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Long large character sums Theses and supervised dissertations / 2019-12
Bujold, Crystel
Abstract
This thesis deals with a central topic in analytic number theory, namely that of characters and more specifically, that of character sums. More precisely, we will develop a result concerning the maximal value that can be attained by some long character sum. In Chapter 1 are discussed the notions and techniques that will be necessary in the elaboration of the proof of the main result. We will discuss notions of harmonic analysis, classical number theoretic techniques, as well as give an overview of smooth numbers. Chapter 2 will serve as an introduction to the theory pertaining to Dirichlet characters and character sums. Basic properties and classical theorems will be covered and we will provide a survey of recent results closely related to the main topic on interest in this thesis. We will give in Chapter 3 a first result which will lead this thesis to diverge into the field of lattices. It comes up as an auxiliary result to the main result, but bares an interest independent to characters. We will discuss the order of magnitude of multiples of a chosen lattice vector, when the multipliers lie in prescribed congruence classes. Chapter 4 will serve as a bridge between lattices and characters and we will study the consequences of applying the theorems we proved in Chapter 3 to characters. We will derive results that will be key to the proof of our main theorem. In Chapter 5, we will prepare the ground for the proof of our main theorem by unveiling some preliminary estimates that will be needed. In particular, the chapter will consist of two parts: one treating of exponential sums, while the other one will be concerned with smooth numbers. Finally, Chapter 6 will be the apex of this thesis and will provide the proof of our main result on character sums. The argument built in this chapter will allow us to prove a lower bound for the maximal value that can be reached by a character among the characters modulo a prime number q.

Anatomy of smooth integers Theses and supervised dissertations / 2017-07
Abstract
The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.

Mean values and correlations of multiplicative functions : the pretentious" approach Theses and supervised dissertations / 2017-07
Klurman, Oleksiy
Abstract
The main theme of this thesis is to study mean values and correlations of multiplicative functions and apply the corresponding results to tackle some open problems. The first chapter contains discussion of several classical facts about mean values of multiplicative functions and statement of the main results of the thesis. The second chapter consists of the article “Mean values of multiplicative functions over the function fields". The main purpose of this chapter is to formulate and prove analog of several classical results due to Wirsing, Hall and Tenenbaum over the function field Fq[x], thus answering questions raised in the recent work of Granville, Harper and Soundararajan. We explain some features of the behaviour of multiplicative functions that are not present in the number field settings. This is accomplished by, among other things, introducing the notion of “pretentiousness" over the function fields. Chapter 3 and Chapter 4 include results of the article “Correlations of multiplicative functions and applications". Here, we give an asymptotic formula for correlations X n_x f1(P1(n))f2(P2(n)) · · · · · fm(Pm(n)) where f . . . ,fm are bounded “pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions f : N ! {−1,+1} with bounded partial sums. This answers a question of Erdos from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either f(n) = ns for Re(s) < 1 or |f(n)| is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of n = a + b where a,b belong to some multiplicative subsets of N. This gives a new "circle method-free" proof of the result of Brüdern. Chapters 5 and Chapter 6 are based on the results obtained in the article “Effective asymptotic formulae for multilinear averages and sign patterns of multiplicative functions," joint with Alexander Mangerel. Using an analytic approach in the spirit of Halász’ mean v value theorem, we compute multidimensional averages x−l X n2[x]l Y 16j6k fj(Lj(n)) as x ! 1, where [x] := [1,x] and L1, . . . ,Lk are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with explicit main and error terms. As an application of our formulae, we establish a local-to-global principle for Gowers norms of multiplicative functions. We reveal and explain irregularities in the distribution of the sign patterns of multiplicative functions by computing the asymptotic densities of the sets of integers n such that a given multiplicative function f : N ! {−1, 1} yields a fixed sign pattern of length 3 or 4 on almost all 3- and 4-term arithmetic progressions, respectively, with first term n. The latter generalizes and refines the work of Buttkewitz and Elsholtz and complements the recent work of Matomaki, Radziwiłł and Tao. We conclude this thesis by discussing some work in progress.

Dénombrement dans les empilements apolloniens généralisés et distribution angulaire dans les extensions quadratiques imaginaires Theses and supervised dissertations / 2015-07
Dias, Dimitri
Abstract
This thesis consists of two main parts. In the first one, we study generalized Apollonian circles and spheres packings. Apollonian packings date back to ancient Greece and, from a number theoretical point of view, are very attractive objects. In this thesis, we will study the set of curvatures (the inverses of the radii) of a generalization of such packings. Under the right conditions, these curvatures are integers. We will show that they satisfy a partial local-global principle, we will count the number of circles of curvatures bounded by some parameter T and we will study the set of prime curvatures. The second part is related to the angular distribution of ideals (or ideal numbers in our case) in imaginary quadratic number fields (which can be seen as the distribution of lattice points on ellipses). We will show that the discrepancy of the set of angles of integral ideal numbers of a given norm is small and we will look at the problem of bounded gaps between prime elements of imaginary quadratic extensions in sectors.

L'Approximation diophantienne simultanée et l'optimisation discrète Theses and supervised dissertations / 2014-12
Rodriguez Caballero, José Manuel
Abstract
Given a (lower or upper) bounded function $f:\mathbb{N}^k \To \Real$ by a mathematical expression. The problem to find the extremal points of $f$ on any bounded set $S \subset \mathbb{N}^k$ is well-defined from a classical point of view. Nevertheless, from a computability theory perspective, it should be avoided the possibility of pathologies when this problem has infinite Kolmogorov complexity. The main constraint is that the order relationship between computable reals is not effectively solvable. We solve this problem by means of a structure containing two algorithms. The first one allows to evaluate the cost function while the second one transforms each value of the cost function in a point in an infinite dimensional vector of a space. We develop three particular cases, one of them corresponding to the Rauzy approximation method. Finally, we make a comparison between the best simultaneous Diophantine approximations obtained by the Rauzy method (our optimization-oriented version of it) and our tetrahedral method, that is one of the main achievement of this work.

Structures linéaires dans les ensembles à faible densité Theses and supervised dissertations / 2014-07
Henriot, Kevin
Abstract
We present three results in additive combinatorics, a recent field at the interface of combinatorics, harmonic analysis and analytic number theory. The unifying theme in our thesis is the detection of additive structure in arithmetic sets of low density, with an emphasis on quantitative aspects. Our first contribution is an improved density estimate for the problem, initiated by Bourgain and others, of finding a long arithmetic progression in a triple sumset. Our second result is a generalization of Sanders' bounds for Roth's theorem from the dense setting, to the setting of small doubling in an arbitrary abelian group. Finally, we extend the best known quantitative results for Roth's theorem in the primes, to all translation-invariant systems of equations of complexity one.

Formes quadratiques ternaires représantant tous les entiers impairs Theses and supervised dissertations / 2013-11
Bujold, Crystel
Abstract
In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers.

On some Density Theorems in Number Theory and Group Theory Theses and supervised dissertations / 2012-08
Abstract
Gowers in his paper on quasirandom groups studies a question of Babai and Sos asking whether there exists a constant $c > 0$ such that every finite group $G$ has a product-free subset of size at least $c|G|$. Answering the question negatively, he proves that for sufficiently large prime $p$, the group $\mathrm{PSL}_2(\mathbb{F}_p)$ has no product-free subset of size $\geq cn^{8/9}$, where $n$ is the order of $\mathrm{PSL}_2(\mathbb{F}_p)$. We will consider the problem for compact groups and in particular for the profinite groups $\SL_k(\mathh{Z}_p)$ and $\Sp_{2k}(\mathbb{Z}_p)$. In Part I of this thesis, we obtain lower and upper exponential bounds for the supremal measure of the product-free sets. The proof involves establishing a lower bound for the dimension of non-trivial representations of the finite groups $\SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Indeed, our theorem extends and simplifies previous work of Landazuri and Seitz, where they consider the minimal degree of representations for Chevalley groups over a finite field. In Part II of this thesis, we move to algebraic number theory. A monogenic polynomial $f$ is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime $q$, using the Chebotarev density theorem, we will show the density of primes $p$, such that $t^q-p$ is monogenic, is greater than or equal to $(q-1)/q$. We will also prove that, when $q=3$, the density of primes $p$, which $\mathbb{Q}(\sqrt[3]{p})$ is non-monogenic, is at least $1/9$.

The distribution of k-tuples of reduced residues Theses and supervised dissertations / 2012-08
Abstract
In 1940 Paul Erdos made a conjecture about the distribution of reduced residues. Here, in this thesis we studied the distribution of k-tuples of reduced residues, and proved extension of Erdos's conjecture for them.

Les progressions arithmétiques dans les nombres entiers Theses and supervised dissertations / 2012-02
Poirier, Antoine
Abstract
The subject of this thesis is the study of arithmetic progressions in the integers. Precisely, we are interested in the size v(N) of the largest subset of the integers from 1 to N that contains no 3 term arithmetic progressions. Therefore, we will construct a large subset of integers with no such progressions, thus giving us a lower bound on v(N). We will begin by looking at the proofs of all the significant lower bounds obtained on v(N), then we will show another proof of the best lower bound known today. For the proof, we will consider points on a large d-dimensional annulus, and count the number of integer points inside that annulus and the number of arithmetic progressions it contains. To obtain bounds on those quantities, it will be interesting to look at the theory behind counting lattice points in high dimensional spheres, which is the subject of the last section.

Sur la répartition des unités dans les corps quadratiques réels Theses and supervised dissertations / 2011-12
Lacasse, Marc-André
Abstract
This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)

Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques Theses and supervised dissertations / 2011-08
Fiorilli, Daniel
Abstract
The main subject of this thesis is the distribution of primes in arithmetic progressions, that is of primes of the form $qn+a$, with $a$ and $q$ fixed, and $n=1,2,3,\dots$ The thesis also compares different arithmetic sequences, according to their behaviour over arithmetic progressions. It is divided in four chapters and contains three articles. The first chapter is an invitation to the subject of analytic number theory, which is followed by a review of the various number-theoretic tools to be used in the following chapters. This introduction also contains some research results, which we found adequate to include. The second chapter consists of the article \emph{Inequities in the Shanks-Rényi prime number race: an asymptotic formula for the densities}, which is joint work with Professor Greg Martin. The goal of this article is to study <>, a phenomenon appearing in <>. Chebyshev was the first to observe that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. More generally, Rubinstein and Sarnak showed the existence of the quantity $\delta(q;a,b)$, which stands for the probability of having more primes of the form $qn+a$ than of the form $qn+b$. In this paper, we establish an asymptotic series for $\delta(q;a,b)$ which is precise to an arbitrary order of precision (in terms of negative powers of $q$). %(it can be instantiated with an error term smaller than any negative power of $q$). We also provide many numerical results supporting our formulas. The third chapter consists of the article \emph{Residue classes containing an unexpected number of primes}. We fix an integer $a \neq 0$ and study the distribution of the primes of the form $qn+a$, on average over $q$. We show that the choice of $a$ has a significant influence on this distribution, and that some arithmetic progressions contain, on average over q, fewer primes than typical arithmetic progressions. This phenomenon is quite surprising since in light of the prime number theorem for arithmetic progressions, the primes are equidistributed in the residue classes $\bmod q$. The fourth chapter consists of the article \emph{The influence of the first term of an arithmetic progression}. In this article we are interested in studying more general arithmetic sequences and finding irregularities similar to those observed in chapter three. Examples of such sequences are the integers which can be written as the sum of two squares, values of binary quadratic forms, prime $k$-tuples and integers free of small prime factors. We show that a broad class of arithmetic sequences exhibits such irregularities over the arithmetic progressions $a\bmod q$, with $a$ fixed and on average over $q$.

Strings of congruent primes in short intervals Theses and supervised dissertations / 2010-11
Freiberg, Tristan
Abstract
Let $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ be the sequence of all primes, and let $q \ge 3$ and $a$ be coprime integers. Recently, and very remarkably, Daniel Shiu proved an old conjecture of Sarvadaman Chowla, which asserts that there are infinitely many pairs of consecutive primes $p_n,p_{n+1}$ for which $p_n \equiv p_{n+1} \equiv a \bmod q$. Now fix a number $\epsilon > 0$, arbitrarily small. In their recent groundbreaking work, Daniel Goldston, J\anos Pintz and Cem Y{\i}ld{\i}r{\i}m proved that there are arbitrarily large $x$ for which the short interval $(x, x + \epsilon\log x]$ contains at least two primes congruent to $a \bmod q$. Given a pair of primes $\equiv a \bmod q$ in such an interval, there might be a prime in-between them that is not $\equiv a \bmod q$. One can deduce that \emph{either} there are arbitrarily large $x$ for which $(x, x + \epsilon\log x]$ contains a prime pair $p_n \equiv p_{n+1} \equiv a \bmod q$, \emph{or} that there are arbitrarily large $x$ for which the $(x, x + \epsilon\log x]$ contains a triple of consecutive primes $p_n,p_{n+1},p_{n+2}$. Both statements are believed to be true, but one can only deduce that one of them is true, and one does not know which one, from the result of Goldston-Pintz-Y{\i}ld{\i}r{\i}m. In Part I of this thesis, we prove that the first of these alternatives is true, thus obtaining a new proof of Chowla's conjecture. The proof combines some of Shiu's ideas with those of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, and so this result may be regarded as an application of their method. We then establish lower bounds for the number of prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n < \epsilon\log p_n$ and $p_{n+1} \le Y$. Assuming a certain unproven hypothesis concerning what is referred to as the level of distribution', $\theta$, of the primes, Goldston-Pintz-Y{\i}ld{\i}r{\i}m were able to prove that $p_{n+1} - p_n \ll_{\theta} 1$ for infinitely many $n$. On the same hypothesis, we prove that there are infinitely many prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n \ll_{q,\theta} 1$. This conditional result is also proved in a quantitative form. In Part II we apply the techniques of Goldston-Pintz-Y{\i}ld{\i}r{\i}m to prove another result, namely that there are infinitely many pairs of distinct primes $p,p'$ such that $(p-1)(p'-1)$ is a perfect square. This is, in a sense, an approximation' to the old conjecture that there are infinitely many primes $p$ such that $p-1$ is a perfect square. In fact we obtain a lower bound for the number of integers $n$, up to $Y$, such that $n = \ell_1\cdots \ell_r$, the $\ell_i$ distinct primes, and $(\ell_1 - 1)\cdots (\ell_r - 1)$ is a perfect $r$th power, for any given $r \ge 2$. We likewise obtain a lower bound for the number of such $n \le Y$ for which $(\ell_1 + 1)\cdots (\ell_r + 1)$ is a perfect $r$th power. Finally, given a finite set $A$ of nonzero integers, we obtain a lower bound for the number of $n \le Y$ for which $\prod_{p \mid n}(p+a)$ is a perfect $r$th power, simultaneously for every $a \in A$.

Sur la distribution des valeurs de la fonction zêta de Riemann et des fonctions L au bord de la bande critque Theses and supervised dissertations / 2009
Lamzouri, Youness
Abstract
Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Distribution of sums of the Legendre symbol Theses and supervised dissertations / 2005
Mehkari, Sana
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

### Selected publications Expand allCollapse all

#### Densité des friables

de La Bretèche, Régis et Granville, Andrew, Densité des friables 142, 303--348 (2014), , Bull. Soc. Math. France

#### Multiplicative functions in arithmetic progressions

Balog, Antal, Granville, Andrew et Soundararajan, Kannan, Multiplicative functions in arithmetic progressions 37, 3--30 (2013), , Ann. Math. Qué.

#### It's as easy as abc''

with Tom Tucker, It's as easy as abc'' , (2012), , Notices of the American Mathematical Society

#### Primitive prime factors in second-order linear recurrence sequences

Granville, Andrew, Primitive prime factors in second-order linear recurrence sequences 155, 431--452 (2012), , Acta Arith.

#### Zeta functions for ideal classes in real quadratic fields, at $s=0$

Biró, Andràs et Granville, Andrew, Zeta functions for ideal classes in real quadratic fields, at $s=0$ 132, 1807--1829 (2012), , J. Number Theory

#### On sharp transitions in making squares

Croot, Ernie, Granville, Andrew, Pemantle, Robin et Tetali, Prasad, On sharp transitions in making squares 175, 1507--1550 (2012), , Ann. of Math. (2)

#### Prime factors of dynamical sequences

Faber, Xander et Granville, Andrew, Prime factors of dynamical sequences 661, 189--214 (2011), , J. Reine Angew. Math.

#### The distribution of the zeros of random trigonometric polynomials

Granville, Andrew et Wigman, Igor, The distribution of the zeros of random trigonometric polynomials 133, 295--357 (2011), , Amer. J. Math.

#### The number of sumsets in a finite field

Alon, Noga, Granville, Andrew et Ubis, Adriàn, The number of sumsets in a finite field 42, 784--794 (2010), , Bull. Lond. Math. Soc.

#### Different approaches to the distribution of primes

Granville, Andrew, Different approaches to the distribution of primes 78, 65--84 (2010), , Milan J. Math.

#### Close lattice points on circles

Cilleruelo, Javier et Granville, Andrew, Close lattice points on circles 61, 1214--1238 (2009), , Canad. J. Math.

#### A good new millennium for the primes

Granville, Andrew, A good new millennium for the primes 12, 547--556 (2009), , Gac. R. Soc. Mat. Esp.

#### Visibility in the plane

Adhikari, Sukumar Das et Granville, Andrew, Visibility in the plane 129, 2335--2345 (2009), , J. Number Theory

#### Pretentiousness in analytic number theory

Granville, Andrew, Pretentiousness in analytic number theory 21, 159--173 (2009), , J. Théor. Nombres Bordeaux

#### Corrigendum to Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis''

Granville, Andrew, Corrigendum to Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis'' 38, 235--237 (2008), , Funct. Approx. Comment. Math.

#### The number of possibilities for random dating

Abrams, Aaron, Canfield, Rod et Granville, Andrew, The number of possibilities for random dating 115, 1265--1271 (2008), , J. Combin. Theory Ser. A

#### Poisson statistics via the Chinese remainder theorem

Granville, Andrew et Kurlberg, Pàr, Poisson statistics via the Chinese remainder theorem 218, 2013--2042 (2008), , Adv. Math.

#### Prime number patterns

Granville, Andrew, Prime number patterns 115, 279--296 (2008), , Amer. Math. Monthly

#### Anatomy of integers

De Koninck, Jean-Marie, Granville, Andrew et Luca, Florian, Anatomy of integers , viii+297 (2008), , American Mathematical Society, Providence, RI

#### Erratum: Prime divisors are Poisson distributed''

Granville, Andrew, Erratum: Prime divisors are Poisson distributed'' 3, 649--651 (2007), , Int. J. Number Theory

#### Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis

Granville, Andrew, Refinements of Goldbach's conjecture, and the generalized Riemann hypothesis 37, 159--173 (2007), , Funct. Approx. Comment. Math.

#### Rational and integral points on quadratic twists of a given hyperelliptic curve

Granville, Andrew, Rational and integral points on quadratic twists of a given hyperelliptic curve Granville, Andrew, Art. ID 027, 24 (2007), , Int. Math. Res. Not. IMRN

#### An uncertainty principle for arithmetic sequences

Granville, Andrew et Soundararajan, K., An uncertainty principle for arithmetic sequences 165, 593--635 (2007), , Ann. of Math. (2)

#### Prime divisors are Poisson distributed

Granville, Andrew, Prime divisors are Poisson distributed 3, 1--18 (2007), , Int. J. Number Theory

#### Large character sums: pretentious characters and the Pólya-Vinogradov theorem

Granville, Andrew et Soundararajan, K., Large character sums: pretentious characters and the Pólya-Vinogradov theorem 20, 357--384 (2007), , J. Amer. Math. Soc.

#### Cycle lengths in a permutation are typically Poisson

Granville, Andrew, Cycle lengths in a permutation are typically Poisson 13, Research Paper 107, 23 (2006), , Electron. J. Combin.

#### Estimates for representation numbers of quadratic forms

Blomer, Valentin et Granville, Andrew, Estimates for representation numbers of quadratic forms 135, 261--302 (2006), , Duke Math. J.

#### Residue races

Granville, Andrew, Shiu, Daniel et Shiu, Peter, Residue races 11, 67--94 (2006), , Ramanujan J.

#### Prime number races

Granville, Andrew et Martin, Greg, Prime number races 113, 1--33 (2006), , Amer. Math. Monthly

#### Selected mathematical reviews

Granville, Andrew, Selected mathematical reviews 43, 93 (2006), , Bull. Amer. Math. Soc. (N.S.)

#### Aurifeuillian factorization

Granville, Andrew et Pleasants, Peter, Aurifeuillian factorization 75, 497--508 (2006), , Math. Comp.

#### Prime number races

Granville, Andrew et Martin, Greg, Prime number races 8, 197--240 (2005), , Gac. R. Soc. Mat. Esp.

#### On the distribution of rational functions along a curve over $\Bbb F_p$ and residue races

Granville, Andrew, Shparlinski, Igor E. et Zaharescu, Alexandru, On the distribution of rational functions along a curve over $\Bbb F_p$ and residue races 112, 216--237 (2005), , J. Number Theory

#### It is easy to determine whether a given integer is prime

Granville, Andrew, It is easy to determine whether a given integer is prime 42, 3--38 (2005), , Bull. Amer. Math. Soc. (N.S.)

#### The square of the Fermat quotient

Granville, Andrew, The square of the Fermat quotient 4, A22, 3 (2004), , Integers

#### The number of unsieved integers up to $x$

Granville, Andrew et Soundararajan, K., The number of unsieved integers up to $x$ 115, 305--328 (2004), , Acta Arith.

#### Errata to: The distribution of values of $L(1,\chi_d)$''

Granville, A. et Soundararajan, K., Errata to: The distribution of values of $L(1,\chi_d)$'' 14, 245--246 (2004), , Geom. Funct. Anal.

#### The distribution of values of $L(1,\chi_d)$

Granville, A. et Soundararajan, K., The distribution of values of $L(1,\chi_d)$ 13, 992--1028 (2003), , Geom. Funct. Anal.

#### Decay of mean values of multiplicative functions

Granville, Andrew et Soundararajan, K., Decay of mean values of multiplicative functions 55, 1191--1230 (2003), , Canad. J. Math.

#### Nombres premiers et chaos quantique

Granville, Andrew, Nombres premiers et chaos quantique Granville, Andrew, 29--44 (2003), , Gaz. Math.

#### The number of fields generated by the square root of values of a given polynomial

Cutter, Pamela, Granville, Andrew et Tucker, Thomas J., The number of fields generated by the square root of values of a given polynomial 46, 71--79 (2003), , Canad. Math. Bull.

#### Unit fractions and the class number of a cyclotomic field

Croot, III, Ernest S. et Granville, Andrew, Unit fractions and the class number of a cyclotomic field 66, 579--591 (2002), , J. London Math. Soc. (2)

#### Upper bounds for $\vert L(1,\chi)\vert$

Granville, Andrew et Soundararajan, K., Upper bounds for $\vert L(1,\chi)\vert$ 53, 265--284 (2002), , Q. J. Math.

#### On the residues of binomial coefficients and their products modulo prime powers

Cai, Tian Xin et Granville, Andrew, On the residues of binomial coefficients and their products modulo prime powers 18, 277--288 (2002), , Acta Math. Sin. (Engl. Ser.)

#### Two contradictory conjectures concerning Carmichael numbers

Granville, Andrew et Pomerance, Carl, Two contradictory conjectures concerning Carmichael numbers 71, 883--908 (2002), , Math. Comp.

#### The spectrum of multiplicative functions

Granville, Andrew et Soundararajan, K., The spectrum of multiplicative functions 153, 407--470 (2001), , Ann. of Math. (2)

#### Large character sums

Granville, Andrew et Soundararajan, K., Large character sums 14, 365--397 (2001), , J. Amer. Math. Soc.

#### Product of integers in an interval, modulo squares

Granville, Andrew et Selfridge, J. L., Product of integers in an interval, modulo squares 8, Research Paper 5, 12 (2001), , Electron. J. Combin.

#### More points than expected on curves over finite field extensions

Brock, Bradley W. et Granville, Andrew, More points than expected on curves over finite field extensions 7, 70--91 (2001), , Finite Fields Appl.

#### Rabinowitsch revisited

Granville, Andrew et Mollin, Richard A., Rabinowitsch revisited 96, 139--153 (2000), , Acta Arith.

#### Zeros of Fekete polynomials

Conrey, B., Granville, A., Poonen, B. et Soundararajan, K., Zeros of Fekete polynomials 50, 865--889 (2000), , Ann. Inst. Fourier (Grenoble)

#### An upper bound on the least inert prime in a real quadratic field

Granville, Andrew, Mollin, R. A. et Williams, H. C., An upper bound on the least inert prime in a real quadratic field 52, 369--380 (2000), , Canad. J. Math.

#### $abc$ implies no Siegel zeros'' for $L$-functions of characters with negative discriminant

Granville, Andrew et Stark, H. M., $abc$ implies no Siegel zeros'' for $L$-functions of characters with negative discriminant 139, 509--523 (2000), , Invent. Math.

#### On the scarcity of powerful binomial coefficients

Granville, Andrew, On the scarcity of powerful binomial coefficients 46, 397--410 (1999), , Mathematika

#### Borwein and Bradley's Apéry-like formulae for $\zeta(4n+3)$

Almkvist, Gert et Granville, Andrew, Borwein and Bradley's Apéry-like formulae for $\zeta(4n+3)$ 8, 197--203 (1999), , Experiment. Math.

#### The set of differences of a given set

Granville, Andrew et Roesler, Friedrich, The set of differences of a given set 106, 338--344 (1999), , Amer. Math. Monthly

#### $ABC$ allows us to count squarefrees

Granville, Andrew, $ABC$ allows us to count squarefrees Granville, Andrew, 991--1009 (1998), , Internat. Math. Res. Notices

#### A binary additive problem of Erdös and the order of $2\bmod p^2$

Granville, Andrew et Soundararajan, K., A binary additive problem of Erdös and the order of $2\bmod p^2$ 2, 283--298 (1998), , Ramanujan J.

#### On the exponential sum over $k$-free numbers

Brüdern, J., Granville, A., Perelli, A., Vaughan, R. C. et Wooley, T. D., On the exponential sum over $k$-free numbers 356, 739--761 (1998), , R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci.

#### Primes at a (somewhat lengthy) glance

Agoh, Takashi, Erdös, Paul et Granville, Andrew, Primes at a (somewhat lengthy) glance 104, 943--945 (1997), , Amer. Math. Monthly

#### Correction to: Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle''

Granville, Andrew, Correction to: Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle'' 104, 848--851 (1997), , Amer. Math. Monthly

#### Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

Granville, Andrew et Ramaré, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients 43, 73--107 (1996), , Mathematika

#### Values of Bernoulli polynomials

Granville, Andrew et Sun, Zhi-Wei, Values of Bernoulli polynomials 172, 117--137 (1996), , Pacific J. Math.

#### Defect zero $p$-blocks for finite simple groups

Granville, Andrew et Ono, Ken, Defect zero $p$-blocks for finite simple groups 348, 331--347 (1996), , Trans. Amer. Math. Soc.

#### On the number of solution of the equation $\sum^n_{i=1}x_i/d_i\equiv 0\pmod 1$, and of diagonal equations in finite fields

Granville, Andrew, Li, Shuguang et Qi, Sun, On the number of solution of the equation $\sum^n_{i=1}x_i/d_i\equiv 0\pmod 1$, and of diagonal equations in finite fields 32, 243--248 (1995), , Sichuan Daxue Xuebao

#### On a problem of Hering concerning orthogonal covers of $\bold K_n$

Granville, A., Gronau, H.-D. O. F. et Mullin, R. C., On a problem of Hering concerning orthogonal covers of $\bold K_n$ 72, 345--350 (1995), , J. Combin. Theory Ser. A

#### On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$

Darmon, Henri et Granville, Andrew, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$ 27, 513--543 (1995), , Bull. London Math. Soc.

#### Harald Cramér and the distribution of prime numbers

Granville, Andrew, Harald Cramér and the distribution of prime numbers Granville, Andrew, 12--28 (1995), , Scand. Actuar. J.

#### On sparse languages $L$ such that $LL=\Sigma^*$

Enflo, Per, Granville, Andrew, Shallit, Jeffrey et Yu, Sheng, On sparse languages $L$ such that $LL=\Sigma^*$ 52, 275--285 (1994), , Discrete Appl. Math.

#### There are infinitely many Carmichael numbers

Alford, W. R., Granville, Andrew et Pomerance, Carl, There are infinitely many Carmichael numbers 139, 703--722 (1994), , Ann. of Math. (2)

#### Integers, without large prime factors, in arithmetic progressions. II

Granville, Andrew, Integers, without large prime factors, in arithmetic progressions. II 345, 349--362 (1993), , Philos. Trans. Roy. Soc. London Ser. A

#### Smoothing smooth'' numbers

Friedlander, John B. et Granville, Andrew, Smoothing smooth'' numbers 345, 339--347 (1993), , Philos. Trans. Roy. Soc. London Ser. A

#### Solution to a problem of Bombieri

Granville, Andrew, Solution to a problem of Bombieri 4, 181--183 (1993), , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.

#### Integers, without large prime factors, in arithmetic progressions. I

Granville, Andrew, Integers, without large prime factors, in arithmetic progressions. I 170, 255--273 (1993), , Acta Math.

#### An upper bound in Goldbach's problem

Deshouillers, Jean-Marc, Granville, Andrew, Narkiewicz, W?adys?aw et Pomerance, Carl, An upper bound in Goldbach's problem 61, 209--213 (1993), , Math. Comp.

#### Computation of the first factor of the class number of cyclotomic fields

Fung, Gilbert, Granville, Andrew et Williams, Hugh C., Computation of the first factor of the class number of cyclotomic fields 42, 297--312 (1992), , J. Number Theory

#### Squares in arithmetic progressions

Bombieri, Enrico, Granville, Andrew et Pintz, Jànos, Squares in arithmetic progressions 66, 369--385 (1992), , Duke Math. J.

#### Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle

Granville, Andrew, Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle 99, 318--331 (1992), , Amer. Math. Monthly

#### Finding integers $k$ for which a given Diophantine equation has no solution in $k$th powers of integers

Granville, Andrew, Finding integers $k$ for which a given Diophantine equation has no solution in $k$th powers of integers 60, 203--212 (1992), , Acta Arith.

#### Limitations to the equi-distribution of primes. III

Friedlander, John et Granville, Andrew, Limitations to the equi-distribution of primes. III 81, 19--32 (1992), , Compositio Math.

#### On a paper of Z. Agur, A. S. Fraenkel and S. T. Klein: The number of fixed points of the majority rule''

Granville, Andrew, On a paper of Z. Agur, A. S. Fraenkel and S. T. Klein: `The number of fixed points of the majority rule'' 94, 147--151 (1991), , Discrete Math.

#### On pairs of coprime integers with no large prime factors

Granville, Andrew, On pairs of coprime integers with no large prime factors 9, 335--350 (1991), , Exposition. Math.

#### Limitations to the equi-distribution of primes. IV

Friedlander, John et Granville, Andrew, Limitations to the equi-distribution of primes. IV 435, 197--204 (1991), , Proc. Roy. Soc. London Ser. A

#### The lattice points of an $n$-dimensional tetrahedron

Granville, Andrew, The lattice points of an $n$-dimensional tetrahedron 41, 234--241 (1991), , Aequationes Math.

#### The prime factors of Wendt's binomial circulant determinant

Fee, Greg et Granville, Andrew, The prime factors of Wendt's binomial circulant determinant 57, 839--848 (1991), , Math. Comp.

#### Subdesigns in Steiner quadruple systems

Granville, Andrew et Hartman, Alan, Subdesigns in Steiner quadruple systems 56, 239--270 (1991), , J. Combin. Theory Ser. A

#### Oscillation theorems for primes in arithmetic progressions and for sifting functions

Friedlander, John, Granville, Andrew, Hildebrand, Adolf et Maier, Helmut, Oscillation theorems for primes in arithmetic progressions and for sifting functions 4, 25--86 (1991), , J. Amer. Math. Soc.

#### A note on sums of primes

Granville, Andrew, A note on sums of primes 33, 452--454 (1990), , Canad. Math. Bull.

#### On the least prime in certain arithmetic progressions

Granville, Andrew et Pomerance, Carl, On the least prime in certain arithmetic progressions 41, 193--200 (1990), , J. London Math. Soc. (2)

#### Bounding the coefficients of a divisor of a given polynomial

Granville, Andrew, Bounding the coefficients of a divisor of a given polynomial 109, 271--277 (1990), , Monatsh. Math.

#### Representing binomial coefficients as sums of squares

Granville, Andrew et Zhu, Yiliang, Representing binomial coefficients as sums of squares 97, 486--493 (1990), , Amer. Math. Monthly

#### On the size of the first factor of the class number of a cyclotomic field

Granville, A., On the size of the first factor of the class number of a cyclotomic field 100, 321--338 (1990), , Invent. Math.

#### Defining Bernoulli polynomials in ${\bf Z}/p{\bf Z}$ (a generic regularity condition)

Granville, Andrew et Shank, H. S., Defining Bernoulli polynomials in ${\bf Z}/p{\bf Z}$ (a generic regularity condition) 108, 637--640 (1990), , Proc. Amer. Math. Soc.

#### On a class of determinants

Granville, Andrew, On a class of determinants 27, 253--256 (1989), , Fibonacci Quart.

#### Limitations to the equi-distribution of primes. I

Friedlander, John et Granville, Andrew, Limitations to the equi-distribution of primes. I 129, 363--382 (1989), , Ann. of Math. (2)

#### On complementary decompositions of the complete graph

Granville, Andrew, Moisiadis, Alexandros et Rees, Rolf, On complementary decompositions of the complete graph 5, 57--61 (1989), , Graphs Combin.

#### Bipartite planes

Granville, Andrew, Moisiadis, Alexandros et Rees, Rolf, Bipartite planes 61, 241--248 (1988), , Congr. Numer.

#### On Sophie Germain type criteria for Fermat's last theorem

Granville, Andrew et Powell, Barry, On Sophie Germain type criteria for Fermat's last theorem 50, 265--277 (1988), , Acta Arith.

#### Nested Steiner $n$-cycle systems and perpendicular arrays

Granville, A., Moisiadis, A. et Rees, R., Nested Steiner $n$-cycle systems and perpendicular arrays 3, 163--167 (1988), , J. Combin. Math. Combin. Comput.

#### The first case of Fermat's last theorem is true for all prime exponents up to $714,591,416,091,389$

Granville, Andrew et Monagan, Michael B., The first case of Fermat's last theorem is true for all prime exponents up to $714,591,416,091,389$ 306, 329--359 (1988), , Trans. Amer. Math. Soc.

#### Sophie Germain's theorem for prime pairs $p,\,6p+1$

Granville, Andrew, Sophie Germain's theorem for prime pairs $p,\,6p+1$ 27, 63--72 (1987), , J. Number Theory

#### On Hajós' conjecture (minimum cycle-partitions of the edge-set of Eulerian graphs)

Granville, Andrew et Moisiadis, Alexandros, On Hajós' conjecture (minimum cycle-partitions of the edge-set of Eulerian graphs) 56, 183--187 (1987), , Congr. Numer.

#### Matrices as the sum of four squares

Granville, Andrew J., Matrices as the sum of four squares 20, 247--251 (1987), , Linear and Multilinear Algebra

#### On Krasner's criteria for the first case of Fermat's last theorem

Granville, Andrew, On Krasner's criteria for the first case of Fermat's last theorem 56, 67--70 (1986), , Manuscripta Math.

#### Powerful numbers and Fermat's last theorem

Granville, Andrew, Powerful numbers and Fermat's last theorem 8, 215--218 (1986), , C. R. Math. Rep. Acad. Sci. Canada

#### Refining the conditions on the Fermat quotient

Granville, Andrew J., Refining the conditions on the Fermat quotient 98, 5--8 (1985), , Math. Proc. Cambridge Philos. Soc.

#### The set of exponents, for which Fermat's last theorem is true, has density one

Granville, Andrew, The set of exponents, for which Fermat's last theorem is true, has density one 7, 55--60 (1985), , C. R. Math. Rep. Acad. Sci. Canada