arXiv subjects

Greg Martin

Publications and source records attributed to Greg Martin.

At least 37 records · Page 2Linked to original sources

arXiv2020

Subproducts of small residue classes

For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by the product of some subset of $\{1,\dots,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue $\pmod p$; we prove that $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result is of intermediate strength between two other results, namely Burthe's proof that the multiplicative group $\pmod p$ is generated by the integers up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$, and Munsch and Shparlinski's result that every reduced residue class $\pmod p$ is represented by the product of some subset of the primes up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$. Unlike the latter result, our proof is elementary and similar in structure to Burgess's proof for the least quadratic nonresidue.

math.NT↗

arXiv2020

A disproof of Hooley's conjecture

Define $G(x;q)$ to be the variance of primes $p\le x$ in the arithmetic progressions modulo $q$, weighted by $\log p$. Hooley conjectured that as soon as $q$ tends to infinity and $x\ge q$, we have the upper bound $G(x;q) \ll x \log q$. In this paper we show that the upper bound does not hold in general, and that $G(x;q)$ can be asymptotically as large as $x (\log q+\log\log\log x)^2/4$.

math.NT↗

arXiv2020

Counting multiplicative groups with prescribed subgroups

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is "maximally non-cyclic", that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $A x/(\log x)^{1-ξ}$ for an explicit constant $A$, where $ξ$ is Artin's constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.

math.NT↗

arXiv2020

Counting Zeros of Dirichlet $L$-Functions

We give explicit upper and lower bounds for $N(T,χ)$, the number of zeros of a Dirichlet $L$-function with character $χ$ and height at most $T$. Suppose that $χ$ has conductor $q>1$, and that $T\geq 5/7$. If $\ell=\log\frac{q(T+2)}{2π}> 1.567$, then \begin{equation*} \left| N(T,χ) - \left( \frac{T}π \log\frac{qT}{2πe} -\frac{χ(-1)}{4}\right) \right| \le 0.22737 \ell + 2 \log(1+\ell) - 0.5. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,χ)|$ for $σ<-1/2$.

math.NT↗

arXiv2020

The smallest invariant factor of the multiplicative group

Let $λ_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the counting function of those integers $n$ for which $λ_1(n)\ne2$. We also give an asymptotic formula, for any even $q\ge4$, for the counting function of those integers $n$ for which $λ_1(n)=q$. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we give an asymptotic formula for the counting function of those integers $n$ all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

math.NT↗

arXiv2019

Exponential sums with reducible polynomials

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $2$ and $3$ and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.

math.NT↗

arXiv2019

Densities in certain three-way prime number races

Let $a_1$, $a_2$, and $a_3$ be distinct reduced residues modulo $q$ satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \pmod q$. We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$, for the logarithmic density of the set of real numbers $x$ for which $π(x;q,a_1) > π(x;q,a_2) > π(x;q,a_3)$. The relationship among the $a_i$ allows us to normalize the error terms for the $π(x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

math.NT↗

arXiv2019

Inclusive prime number races

Let $π(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $π(x;q,a_1) > π(x;q,a_2) > \cdots > π(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Rényi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, and if the logarithmic density of the set of such $x$ exists and is positive, the prime number race is called inclusive. In breakthrough research, Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can assume that almost all of the zeros may be involved in $\mathbb{Q}$-linear relations; and we can also conclude more strongly that the associated limiting distribution has mass everywhere. This work makes use of a number of ideas from probability, the explicit formula from number theory, and the Kronecker-Weyl equidistribution theorem.

math.NT↗

arXiv2019

Primitive points in rational polygons

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{π^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $Ω_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $ϕ(n)$.

math.NT↗

arXiv2019

Primes in prime number races

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $ζ(s)$, that the set of real numbers $x\ge2$ for which $π(x)>$ li$(x)$ has a logarithmic density, which they computed to be about $2.6\times10^{-7}$. A natural problem is to examine the actual primes in this race. We prove, assuming RH and LI, that the logarithmic density of the set of primes $p$ for which $π(p)>$ li$(p)$ relative to the prime numbers exists and is the same as the Rubinstein-Sarnak density. We also extend such results to a broad class of prime number races, including the "Mertens race" between $\prod_{p< x}(1-1/p)^{-1}$ and $e^γ\log x$ and the "Zhang race" between $\sum_{p\ge x}1/(p\log p)$ and $1/\log x$. These latter results resolve a question of the first and third author from a previous paper, leading to further progress on a 1988 conjecture of Erdős on primitive sets.

math.NT↗

arXiv2018

Explicit bounds for primes in arithmetic progressions

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $θ(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| θ(x; q, a) - \frac{x}{ϕ(q)} \bigg| < \frac1{160} \frac{x}{\log x}, $$ for all $x \ge 8 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $π(x;q,a)$ and $ψ(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod q$ when $q\le1200$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,χ)$ for quadratic characters $χ$, and an improved explicit upper bound for exceptional zeros.

math.NT↗

arXiv2018

The distribution of sums and products of additive functions

The celebrated Erdős--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if $g_1(n), \ldots, g_k(n)$ is a collection of functions satisfying certain mild hypotheses for which an Erdős--Kac-type normal distribution law holds, and if $Q(x_1, \ldots, x_k)$ is a polynomial with nonnegative real coefficients, then $Q(g_1(n), \ldots, g_k(n))$ also obeys a normal distribution law. We also show that a similar result can be obtained if the set of inputs $n$ is restricted to certain subsets of the natural numbers, such as shifted primes. Our proof uses the method of moments. We conclude by providing examples of our theorem in action.

math.NT↗

arXiv2018

Factorization tests and algorithms arising from counting modular forms and automorphic representations

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $Γ_0(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $Γ_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.

math.NT↗

arXiv2017

Smooth values of polynomials

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in\mathbb{Z}[t]$ and any $ε> 0$, there are infinitely many $n\in\mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^ε$.

math.NT↗

arXiv2017

The distribution of the number of subgroups of the multiplicative group

Let $I(n)$ denote the number of isomorphism classes of subgroups of $(\Bbb Z/n\Bbb Z)^\times$, and let $G(n)$ denote the number of subgroups of $(\Bbb Z/n\Bbb Z)^\times$ counted as sets (not up to isomorphism). We prove that both $\log G(n)$ and $\log I(n)$ satisfy Erdös-Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that $\log G(n)$ is not an additive function but is closely related to the sum of squares of additive functions. We also establish the orders of magnitude of the maximal orders of $\log G(n)$ and $\log I(n)$.

math.NT↗

arXiv2015

An annotated bibliography of work related to gender in science

The purpose of this manuscript is to gather together a large amount of source material pertaining to women in mathematics, from studies of girls in elementary school through data on females winning prizes for mathematical research. Along the way, we have also gathered a large amount of material from the psychology and sociology literature on implicit biases more generally, particularly pertaining to gender. This source material was then used to support the writing of the article "Addressing the underrepresentation of women in mathematics conferences". We have referred to primary research literature whenever possible, although we have also included well-written blog posts, organizational web sites, self-published articles by research organizations, and even a YouTube video. Each bibliography entry is accompanied by some remarks summarizing its content and representative quotes from the articles themselves. Much of the work in this bibliography contains a large number of further references to the relevant research literature.

math.HO↗

arXiv2015

Addressing the underrepresentation of women in mathematics conferences

Despite significant improvements over the last few generations, the discipline of mathematics still counts a disproportionately small number of women among its practitioners. These women are underrepresented as conference speakers, even more so than the underrepresentation of women among PhD-earners as a whole. This underrepresentation is the result of implicit biases present within all of us, which cause us (on average) to perceive and treat women and men differently and unfairly. These mutually reinforcing biases begin in primary school, remain active through university study, and continue to oppose women's careers through their effects on hiring, evaluation, awarding of prizes, and inclusion in journal editorial boards and conference organization committees. Underrepresentation of women as conference speakers is a symptom of these biases, but it also serves to perpetuate them; therefore, addressing the inequity at conferences is valuable and necessary for countering this underrepresentation. We describe in detail the biases against women in mathematics, knowing that greater awareness of them leads to a better ability to mitigate them. Finally, we make explicit suggestions for organizing conferences in ways that are equitable for female mathematicians.

math.HO↗

arXiv2014

abc triples

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than c. Triples of numbers satisfying a+b=c are called abc triples if the product of their distinct prime divisors is strictly less than c. We catalog what is known about abc triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the $abc$ conjecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture.

math.NT↗