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Elliptic curves and modular forms /Courbes elliptiques et formes modulaires
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prize lecture(December 2021)

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Recent Preprints
It is not “Bézout’s identity”
Given two non-zero integers a and b there exist integers m and n for whicham − bn = gcd( a, b).
An increasing number of mathematicians have been calling this “Bézout’s identity” (or lemmaor theorem’) some encouraged by finding “identité de Bézout” in Bourbaki’s Éléments demathématique.
However (we will explain that) this is in Euclid’s work from around 300 B.C., when hiswritings are interpreted in context. So why does he not get credit? Some authors learned thename “Bézout’s identity” and copied the misattribution. Others, like some Nicolas Bourbakicollaborators, have perhaps browsed Euclid’s results, but in a form written for the modern
mathematician, and missed out on what he really did (though certainly others, such as Weil, didnot). In this article we will carefully explain what Euclid’s arguments are and what his approach
was. We will also share Emmanuel Kowalski’s guess as to the reasons behind Bourbaki’s
misnomer.
To appreciate Euclid, you need to read his work in appropriate context: Lengths were thecentral object of study to the geometer Euclid, though he brilliantly developed the theory of
the numbers that measured those lengths. Today’s mathematicians read his number theory
results as being about abstract numbers not measurements. However the correct interpretation
changes how these results are perceived; moreover Euclid’s proofs make clear Euclid’s intentions.
These misperceptions reflect recent discussions about the difficulties sometimes faced byindigenous people when learning mathematics. We will discuss how some indigenous groups
may learn numbers in certain practical contexts, not as abstract entities, and struggle whencurricula assume that we all share abstract numbers as a basic, primary fully-absorbed working
tool.
——————
Improved stability for the size and structure of sumsets(with Jack Smith and Aled Walker )Let A ⊂ Zd be a finite set. It is known that the sumset N A has predictable size (| N A| =PA ( N ) for some PA ( X ) ∈ Q[ X ]) and structure (all of the lattice points in some finite cone
other than all of the lattice points in a finite collection of exceptional subcones), once N
is larger than some threshold. In previous work, the first effective bounds for both of thesethresholds were established, for an arbitrary set A. In this article we substantially improveeach of these bounds, coming much closer to the corresponding lower bounds known.
——————The multiplication table constant and sums of two squares,(with Cihan Sabuncu and Alisa Sedunova), Acta Arihtmetica
We will show that the number of integers ≤ x that can be written as the square of an integer
plus the square of a prime equals π2 · logx x minus a secondary term of size x/(log x )1+δ+o(1) ,1+log log 2

where δ := 1 − log 2 = 0.0860713320 . . . is the multiplication table constant. Detailed
heuristics suggest that this secondary term is asymptotic topx
1·log log x (log x )1+δ
log log xtimes a bounded, positive, 1-periodic, non-constant function of log 2 .
——————
Missing digits, and good approximations, Bulletin of the AMSJames Maynard has taken the analytic number theory world by storm in the last decade,proving several important and surprising theorems, resolving questions that had seemed farout of reach. He is perhaps best known for his work on small and large gaps between primes.
In this article we will discuss two other Maynard breakthroughs:— Mersenne numbers take the form 2n − 1 and so appear as 111 . . . 111 in base 2, having nodigit ‘0’. It is a famous conjecture that there are infinitely many such primes. More generally
it was, until Maynard’s work, an open question as to whether there are infinitely many primes
that miss any given digit, in any given base. We will discuss Maynard’s beautiful ideas thatwent into partly resolving this question.
— In 1926, Khinchin gave remarkable conditions for when real numbers can usually be “wellapproximated” by infinitely many rationals. However Khinchin’s theorem regarded 1/2, 2/4,3/6 as distinct rationals and so could not be easily modified to cope, say, with approximations
by fractions with prime denominators. In 1941 Duffin and Schaefer proposed an appropriate
but significantly more general analogy involving approximation only by reduced fractions (whichis much more useful). We will discuss its recent resolution by Maynard together with Dimitris
Koukoulopoulos.
——————
Fibonacci primes, primes of the form 2n − k and beyond,(with Jon Grantham), Journal of Number Theory
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences(un )n≥0 in the integers. By tweaking a heuristic which is successfully used to predict thenumber of prime values of polynomials, we guess that either there are only finitely many primes
un , or else there exists a constant cu > 0 (which we can give good approximations to) suchthat there are ∼ cu log N primes un with n ≤ N, as N → ∞. We compare our conjecture to
the limited amount of data that we can compile.
——————
Proof in the time of machines, Bulletin of the AMS
We compare the values associated with (traditional) community based proof verification to
those associated with computer proof verification. We propose ways that computer proofs
might incorporate successful strategies from human experiences.
——————Classifying linear division sequences, submitted
We classify all linear division sequences in the integers, a problem going back to at least the1930s. As a corollary we also classify those linear recurrence sequences in the integers for which( xm , xn ) = ± x(m,n) . We also show that if two linear division sequences have a large common
factor infinitely often then they are each divisible by a common linear division sequence onsome arithmetic progression. Moreover our proofs also work for polynomials. The key to ourproofs are Ritt’s irreducibility theorem and the subspace theorem (of Schmidt and Schlickewei),in a direction developed by Bugeaud, Corvaja and Zannier.
——————Do proofs yield objective truth, or are they culturally robust at best?, Annals of Mathematics
and Philosophy
In this article we explore what is accepted in the mathematical community as proof, how wedeal with the Godel “crisis”, is there an objective standard for fully justified proof, how does
computer proof change all that, etc.
——————
Three conjectures about character sums
(with Alexander (Sacha) Mangerel), Mathematische Zeitschrift
We establish that three well-known and rather different looking conjectures about Dirichlet
characters and their (weighted) sums, (concerning the Polya-Vinogradov theorem for maximalcharacter sums, the maximal admissible range in Burgess’ estimate for short character sums,
and upper bounds for L(1, χ) and L(1 + it, χ)) are more-or-less “equivalent”. We also obtaina new mean value theorem for logarithmically weighted sums of 1-bounded multiplicativefunctions.
——————
Consecutive real quadratic fields with large class numbers(with Giacomo Cherubini, Alessandro Fazzari, Vítězslav Kala, and Pavlo Yatsynal),
1/2−o (1) integers
IMRN ) For a given positive integer k, we prove√ that there are√ at least xd ≤ x such that the real quadratic fields Q( d + 1), . . . , Q( d + k) have class numbers√ log log dessentially as large as possible, that is k d log d——————The man who loved problems: Richard K Guy(with Carl Pomerance),
Notices of the American Mathematical Society
A celebration of the life of one of the most colorful figures in contemporary mathematics.
Richard Guy, popular author, columnist, educator, researcher, and avid mountain climber,passed away in 2020 at the age of 103.
——————
In mathematics, as in art,
Journal of Humanistic Mathematics (to appear)
We compare appreciation of mathematics with appreciation of impressionist art

——————
Effective results on the size and structure of sumsets(with George Shakan and Aled Walker ), submitted
Let A ⊂ Zd be a finite set. It is known that N A has a particular size (| N A| = PA ( N ) forsome PA ( X ) ∈ Q[ X ] ) and structure (all of the lattice points in a cone other than certain
exceptional sets), once N is larger than some threshold. In this article we give the first effective
upper bounds for this threshold for arbitrary A . Such explicit results were only previously
known in the special cases when d = 1 , when the convex hull of A is a simplex or when
| A| = d + 2 , results which we improve.
——————
Large deviations of sums of random variables(with Youness Lamzouri)
Lithuanian Mathematics Journal 61 (2021), 345–372
In this paper, we investigate the large deviations of sums of weighted random variables that areapproximately independent, generalizing and improving some of the results of Montgomery andOdlyzko. We are motivated by examples arising from number theory, including the sequences
pit , χ( p), χd ( p), λ f ( p), and Klq ( a − n, b); where p ranges over the primes, t varies in a large
interval, χ varies among all characters modulo q, χd varies over quadratic characters attachedto fundamental discriminants |d| ≤ x, λ f (n) are the Fourier coefficients of holomorphic cusp
forms f of (a large) weight k for the full modular group, and Klq ( a, b) are the normalized
Kloosterman sums modulo a large prime q, where a, b vary in (Fq )× .
——————
Exponential sums with multiplicative coefficients and applications(with Régis de la Bréteche)
Transactions of the American Mathematical Society (to appear)
We show that if an exponential sum with multiplicative coefficients is large then the associatedmultiplicative function is “pretentious”. This leads to applications in the circle method, and anatural interpretation of the local-global principle.
——————
Sieving intervals and Siegel zeros,
Acta Arithmetica (to appear)
Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)JurkatRichert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec’s lower
bound on Jacobsthal’s problem, as well as minor improvements to the Brun-Titchmarsh
Theorem. We also deduce an improved (though conditional) lower bound on the longest gaps
between primes, and rework Cramér’s heuristic in this situation to show that we would expect
gaps around x that are significantly larger than (log x )2 .
——————
Primes in short intervals: Heuristics and calculations(with Allysa Lumley )Experimental Mathematics (online 1/7/21)
We formulate, using heuristic reasoning, precise conjectures for the range of the number ofprimes in intervals of length y around x, where y  (log x )2 . In particular we conjecture thatthe maximum grows surprisingly slowly as y ranges from log x to (log x )2 . We will show that
our conjectures are somewhat supported by available data, though not so well that there may
not be room for some modification.
——————
A tight structure theorem for sumsets(with Aled Walker )Proceedings of the American Mathematical Society 149 (2021), 4073–4082
Let A = {0 = a0 < a1 < · · · < a`+1 = b} be a finite set of non-negative integers. We provethat the sumset N A has a certain easily-described structure, provided that N ≥ b − `, asrecently conjectured by Granville and Shakan. We also classify those sets A for which thisbound cannot be improved.
——————The Frobenius postage stamp problem, and beyond(with George Shakan)
Acta Mathematica Hungarica 161 (2020), 700–718.
Let A be a finite subset of Zn , which generates Zn additively. We provide a precise descriptionof the N-fold sumsets N A for N sufficiently large, with some explicit bounds on “sufficientlylarge." For example if A has exactly three elements we provide a precise description of N A forall N ≥ 1.

Number Theory Journals
Acta Arithmetica
Algebra and Number Theory
Integers: Elect. J. Number theory
International J. Number theory
J. Number theory
J. Théorie de Nombres, Bordeaux
Ramanujan Journal
Research in Number Theory

Number theory websites
Number Theory Web
Number Theory listserve
Dickson’s History: vol 1, vol 2, and vol 3.
Paul Erdos’s papers
L-functions / Modular Forms database
Online encyclopaedia of integer sequences
Women in Number Theory

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