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Andrew Granville

Publications and source records attributed to Andrew Granville.

At least 37 records · Page 2Linked to original sources
arXiv2021

Large deviations of sums of random variables

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 $p^{it}$, $χ(p)$, $χ_d(p)$, $λ_f(p)$, and $\text{Kl}_q(a-n, b)$; where $p$ ranges over the primes, $t$ varies in a large interval, $χ$ varies among all characters modulo $q$, $χ_d$ varies over quadratic characters attached to fundamental discriminants $|d|\leq x$, $λ_f(n)$ 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 $(\mathbb{F}_q)^{\times}$.

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arXiv2021

Primes in short intervals: Heuristics and calculations

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.

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arXiv2021

A tight structure theorem for sumsets

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 Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved.

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arXiv2020

Sieving intervals and Siegel zeros

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 Cramér's heuristic in this situation to show that we would expect gaps around $x$ that are significantly larger than $(\log x)^2$.

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arXiv2020

The Frobenius postage stamp problem, and beyond

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."

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arXiv2020

Sieve weights and their smoothings

We obtain asymptotic formulas for the $2k$th moments of partially smoothed divisor sums of the Möbius 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".

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arXiv2019

Bombieri-Vinogradov for multiplicative functions, and beyond the $x^{1/2}$-barrier

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions 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 find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class $a$ we extend such averages out to moduli $\leq x^{\frac {20}{39}-δ}$.

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arXiv2018

Natural exact covering systems and the reversion of the Möbius series

We prove that the number of natural exact covering systems of cardinality $k$ is equal to the coefficient of $x^k$ in the reversion of the power series $\sum_{k \ge 1} μ(k) x^k$, where $μ(k)$ is the usual number-theoretic Möbius function. Using this result, we deduce an asymptotic expression for the number of such systems.

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arXiv2018

Beyond the LSD method for the partial sums of multiplicative functions

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $α$ on average. In the literature, the average is usually taken to be $α$ 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.

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arXiv2017

Squares in arithmetic progressions and infinitely many primes

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.

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arXiv2017

Using Dynamical Systems to Construct Infinitely Many Primes

Euclid's proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. After acceptance Soundararajan noted the beautiful and fast converging formula: $$ τ= a^{1/(d-1)} x_0 \cdot \lim_{n\to \infty} \prod_{m=1}^n \left(\frac{x_m}{ax_{m-1}^d} \right)^{1/d^m} $$

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arXiv2017

Large character sums: Burgess's theorem and zeros of $L$-functions

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

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arXiv2017

The frequency and the structure of large character sums

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

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arXiv2017

A more intuitive proof of a sharp version of Halász's theorem

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 $α$" 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.

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