Advert at Prime Suspects: The anatomy of integers and permutations (Released: Aug 2019)

(with George Shakan and Aled Walker)

Let \( A \subset \mathbb{Z}^d\) be a finite set. It is known that \( NA\) has a particular size (\( \vert NA\vert = P_A(N)\) for some \( P_A(X) \in \mathbb{Q}[X]\) )
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 \( \vert A\vert = d+2\) , results which we improve.

(with Youness Lamzouri)

In this paper, we investigate the large deviations of sums of weighted random variables that are * approximately independent*, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences \( p^{it}, \chi(p), \chi_d(p), \lambda_f(p)\), and \( \text{Kl}_q(a-n, b)\); where \( p\) ranges over the primes, \( t\) varies in a large interval, \( \chi\) varies among all characters modulo \( q\), \( \chi_d\) varies over quadratic characters attached to fundamental discriminants \( |d|\leq x\), \( \lambda_f(n)\) are the Fourier coefficients of holomorphic cusp forms \( f\) of (a large) weight \( k\) for the full modular group, and \( \text{Kl}_q(a, b)\) are the normalized Kloosterman sums modulo a large prime \( q\), where \( a, b\) vary in \( (\mathbb{F}_q)^{\times}\).

(with Regis de la Breteche)

We show that if an exponential sum with multiplicative
coefficients is large then the associated multiplicative function is
``pretentious''. This leads to applications in the circle method, and a natural interpretation of the local-global principle.

Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert 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 Cram\'er's heuristic in this situation to show that we would expect gaps around \( x\) that are significantly larger than \( (\log x)^2.\)

(with Allysa Lumley)

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length \( y\) around \( x\), where \( y\ll (\log x)^2\).
In particular we conjecture that the 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.

(with Aled Walker)

Let \( A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\} \) be a finite set of non-negative integers. We prove that the sumset \( NA \) has a certain easily-described structure, provided that \( N \geqslant b-\ell \), as recently conjectured by Granville and Shakan. We also classify those sets \( A \) for which this bound cannot be improved.

(with George Shakan)

Let \(A\) be a finite subset of \(\mathbb{Z}^n\), which generates \(\mathbb{Z}^n\) additively. We provide a precise description of the \(N\)-fold sumsets \(NA\) for \(N\) sufficiently large, with some explicit bounds on ``sufficiently large." For example if \( A\) has exactly three elements we provide a precise description of \( NA\) for all \( N\geq 1\).

(with Dimitris Koukoulopoulos)

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function \( f\) whose prime values are \(\alpha\) on average. In the literature, the average is usually taken to be \(\alpha\) with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of \(f\) , developing new techniques to do so.

(with Xuancheng (Fernando) Shao)

Let \( f\) and \( g\) be \( 1\)-bounded multiplicative functions for which \( f*g=1_{.=1}\). The Bombieri-Vinogradov Theorem holds for both \( f\) and \( g\)
if and only if
the Siegel-Walfisz criterion holds for both \( f\) and \( g\), and
the Bombieri-Vinogradov Theorem holds for \( f\) restricted to the primes.

(with Adam Harper and K. Soundararajan)

Halász's Theorem gives an upper bound for the mean value of a multiplicative function \( f\). The bound is sharp for general such \( f\), and, in particular, it implies that a multiplicative function with \( |f(n)|\le 1\) has either mean value \( 0\), or is ``close to" \( n^{it}\) for some fixed \( t\). The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's Theorem).

(with Adam Harper and K. Soundararajan)

We prove a sharp version of Halász's theorem on sums \( \sum_{n \leq x} f(n)\) of multiplicative functions \( f\) with \( |f(n)|\le 1\). Our proof avoids the ``average of averages'' and ``integration over \( \alpha\)'' manoeuvres that are present in many of the existing arguments. Instead, motivated by the circle method we express \( \sum_{n \leq x} f(n)\) as a triple Dirichlet convolution, and apply Perron's formula.

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics.

(with Sary Drappeau and Xuancheng (Fernando) Shao)

We show that smooth-supported multiplicative functions \( f\) are well-distributed in arithmetic progressions \( a_1a_2^{-1} \pmod q\) on average over moduli \( q\leq x^{3/5-\varepsilon}\) with \( (q,a_1a_2)=1\) . This extends our results in
* Bombieri-Vinogradov for multiplicative functions, and beyond the \( x^{1/2}\)-barrier * (with Fernando Shao).

(with Xuancheng (Fernando) Shao)

Part-and-parcel of the study of ``multiplicative number theory'' is the study of the distribution of multiplicative functions \( f\) in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we prove that such a result has been so elusive because \( f \) can be ``pretentious'' in two different ways. Firstly it might correlate with a character of small conductor, which can be ruled out by assuming a ``Siegel-Walfisz'' type criterion for \( f\). Secondly \( f\) might be particularly awkward on large primes, and this can be avoided by restricting our attention to smoothly supported \( f\). Under these assumptions we recover a Bombieri-Vingradov Theorem for multiplicative \( f\).
For a fixed residue class \( a\) we extend such averages out to moduli \( \leq x^{\frac {20}{39}-\delta}\) .

(with Dimitris Koukoulopoulos and James Maynard)

We obtain asymptotic formulas for the \( 2k\)th moments of partially smoothed divisor sums of the M\"obius function.
When \( 2k\) is small compared with \( A\), the level of smoothing, then the main contribution to the moments come from integers with only large prime factors, as one would hope for in sieve weights. However if \( 2k\) is any larger, compared with \( A\), then the main contribution to the moments come from integers with quite a few prime factors, which is not the intention when designing sieve weights. The threshold for ``small'' occurs when \( A=\frac 1{2k} \binom{2k}{k}-1\).
One can ask analogous questions for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, with even less cancellation in the divisor sums. We give, we hope, a plausible explanation for this phenomenon, by studying the analogous sums for Dirichlet characters, and obtaining each type of behaviour depending on whether or not the character is ``exceptional''.

(with K. Soundararajan)

We study the conjecture that \( \sum_{n\leq x} \chi(n)=o(x)\) for any primitive Dirichlet character \( \chi \pmod q\) with \( x\geq q^\epsilon\), which is known to be true if the Riemann Hypothesis holds for \( L(s,\chi)\). We show that it holds under the weaker assumption that `\(100\%\)' of the zeros of \( L(s,\chi)\) up to height \( \tfrac 14\) lie on the
critical line. We also establish various other consequences of having large character sums; for example, that if the conjecture holds for \( \chi^2\) then it also holds for \( \chi\).

In this Monthly note, we use Van der Waerden's Theorem and Fermat's Theorem on four squares in an arithmetic progression to prove that there are infinitely many primes.

(with Igor Wigman)

We study the small scale distribution of the L^2-mass of
eigenfunctions of the Laplacian on the the two-dimensional flat torus. Given an orthonormal
basis of eigenfunctions, Lester and Rudnick showed the existence of a density one subsequence
whose L^2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L^2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale.
This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.

(with Jonathan Bober, Leo Goldmakher and Dimitris Koukoulopoulos)

Let \( M(\chi)\) denote the maximum of \( |\sum_{n\le N}\chi(n)|\) for a given non-principal Dirichlet character \( \chi \pmod q\), and let \( N_\chi\) denote a point at which the maximum is attained. In this article we study the distribution of \( M(\chi)/\sqrt{q}\) as one varies over characters \( \pmod q\), where \( q\) is prime, and investigate the location of \( N_\chi\). We show that the distribution of \( M(\chi)/\sqrt{q}\) converges weakly to a universal distribution \( \Phi\), uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for \( \Phi\)'s tail. Almost all \( \chi\) for which \( M(\chi)\) is large are odd characters that are \( 1\)-pretentious. Now, \( M(\chi)\ge |\sum_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)|\), and one knows how often the latter expression is large, which has been how earlier lower bounds on \( \Phi\) were mostly proved. We show, though, that for most \( \chi\) with \( M(\chi)\) large, \( N_\chi\) is bounded away from \( q/2\), and the value of \( M(\chi)\) is little bit larger than \( \frac{\sqrt{q}}{\pi} |L(1,\chi)|\).

(with Adam Harper and K. Soundararajan)

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields.
We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.

(with David Dummit and Hershy Kisilevsky)

David Dummit and Hershy Kisilevsky observed from calculation that the Legendre symbols \( (p/q) \) and \( (q/p) \) are unequal for rather more than a quarter of the pairs of odd primes \(p\) and \(q\) with \(pq\leq x\), during some calculations. In fact almost \( 30 \% \)
of the \(pq\)'s up to a million satisfy \( p\equiv q\equiv 3 \pmod 4\). Together we found that this is no accident and that the bias up to \(x\) is roughly \( 1 +1/3(\log\log x-1)\). This is a much stronger bias than the traditional "prime race" problem. When doing the math one finds that this problems about \(pq\)'s is equivalent to the prime race problem, for primes \(=3 \pmod 4\) versus those \(=1 \pmod 4\), in which we weight each prime by its reciprocal.