arXiv subjects

D. H. J. Polymath

Publications and source records attributed to D. H. J. Polymath.

9 recordsLinked to original sources

arXiv2025

The probability that a random triple of dice is transitive

An $n$-sided die is an $n$-tuple of positive integers. We say that a die $(a_1,\dots,a_n)$ beats a die $(b_1,\dots,b_n)$ if the number of pairs $(i,j)$ such that $a_i>b_j$ is greater than the number of pairs $(i,j)$ such that $a_i<b_j$. We show that for a natural model of random $n$-sided dice, if $A, B$ and $C$ are three random dice then the probability that $A$ beats $C$ given that $A$ beats $B$ and $B$ beats $C$ is approximately 1/2. In other words, the information that $A$ beats $B$ and $B$ beats $C$ has almost no effect on the probability that $A$ beats $C$. This proves a statement that was conjectured by Conrey, Gabbard, Grant, Liu and Morrison for a different model.

math.CO↗

arXiv2019

Effective approximation of heat flow evolution of the Riemann $ξ$ function, and a new upper bound for the de Bruijn-Newman constant

For each $t \in \mathbf{R}$, define the entire function $$ H_t(z) := \int_0^\infty e^{tu^2} Φ(u) \cos(zu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ This is essentially the heat flow evolution of the Riemann $ξ$ function. From the work of de Bruijn and Newman, there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is equivalent to the assertion $Λ\leq 0$; recently, Rodgers and Tao established the matching lower bound $Λ\geq 0$. Ki, Kim and Lee established the upper bound $Λ< \frac{1}{2}$. In this paper we establish several effective estimates on $H_t(x+iy)$ for $t \geq 0$, including some that are accurate for small or medium values of $x$. By combining these estimates with numerical computations, we are able to obtain a new upper bound $Λ\leq 0.22$ unconditionally, as well as improvements conditional on further numerical verification of the Riemann hypothesis. We also obtain some new estimates controlling the asymptotic behavior of zeroes of $H_t(x+iy)$ as $x \to \infty$.

math.NT↗

arXiv2018

Homogeneous length functions on groups

A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which saturate the triangle inequality whenever $x=y$, or equivalently those that are homogeneous in the sense that $\ell(x^n) = n\,\ell(x)$ for all $n\in\mathbb{N}$. We show that this implies that $\ell([x,y])=0$ for all $x,y \in G$. This leads to a classification of such pseudo-length functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property.

math.GR↗

arXiv2014

Variants of the Selberg sieve, and bounded intervals containing many primes

For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $H_1 \leq 4680$, and then by Maynard to $H_1 \leq 600$, who also established for the first time a finiteness result for $H_m$ for $m \geq 2$, and specifically that $H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $H_1 \leq 12$, improving upon the previous bound $H_1 \leq 16$ of Goldston, Pintz, and Yıldırım, as well as the bound $H_m \ll m^3 e^{2m}$. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound $H_1 \leq 246$ unconditionally, and $H_1 \leq 6$ under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple $(h_1,h_2,h_3)$, there are infinitely many $n$ for which at least two of $n+h_1,n+h_2,n+h_3$ are prime. We modify the "parity problem" argument of Selberg to show that this result is the best possible that one can obtain from purely sieve-theoretic considerations. For larger $m$, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound $H_m \ll m e^{(4-\frac{24}{181})m}$, or $H_m \ll m e^{2m}$ under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for $H_m$ when $m=2,3,4,5$.

math.NT↗

arXiv2014

The "bounded gaps between primes" Polymath project - a retrospective

For any $m \geq 1$, let $H_m$ denote the quantity $H_m := \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ denotes the $n^{\operatorname{th}}$ prime; thus for instance the twin prime conjecture is equivalent to the assertion that $H_1$ is equal to two. In a recent breakthrough paper of Zhang, a finite upper bound was obtained for the first time on $H_1$; more specifically, Zhang showed that $H_1 \leq 70000000$. Almost immediately after the appearance of Zhang's paper, improvements to the upper bound on $H_1$ were made. In order to pool together these various efforts, a \emph{Polymath project} was formed to collectively examine all aspects of Zhang's arguments, and to optimize the resulting bound on $H_1$ as much as possible. After several months of intensive activity, conducted online in blogs and wiki pages, the upper bound was improved to $H_1 \leq 4680$. As these results were being written up, a further breakthrough was introduced by Maynard, who found a simpler sieve-theoretic argument that gave the improved bound $H_1 \leq 600$, and also showed for the first time that $H_m$ was finite for all $m$. The polymath project, now with Maynard's assistance, then began work on improving these bounds, eventually obtaining the bound $H_1 \leq 246$, as well as a number of additional results, both conditional and unconditional, on $H_m$. In this article, we collect the perspectives of several of the participants to these Polymath projects, in order to form a case study of online collaborative mathematical activity, and to speculate on the suitability of such an online model for other mathematical research projects.

math.HO↗

arXiv2014

New equidistribution estimates of Zhang type

We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} + \frac{7}{300}$.

math.NT↗

arXiv2012

Deterministic methods to find primes

Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by selecting numbers between $N$ and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one seeks a deterministic method, then the problem is much more difficult, unless one assumes some unproven conjectures in number theory; brute force methods give a $O(N^{1+o(1)})$ algorithm, and the best unconditional algorithm, due to Odlyzko, has a run time of $O(N^{1/2 + o(1)})$. In this paper we discuss an approach that may improve upon the $O(N^{1/2+o(1)})$ bound, by suggesting a strategy to determine in time $O(N^{1/2-c})$ for some $c>0$ whether a given interval in $[N,2N]$ contains a prime. While this strategy has not been fully implemented, it can be used to establish partial results, such as being able to determine the \emph{parity} of the number of primes in a given interval in $[N,2N]$ in time $O(N^{1/2-c})$.

math.NT↗

arXiv2010

Density Hales-Jewett and Moser numbers

For any $n \geq 0$ and $k \geq 1$, the \emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,...,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$); this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established. Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c'_{n,k}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,...,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,...,6$ is $1,2,6,16,43,124,353$. We also prove some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$, and also establishing the asymptotic lower bound $c_{n,k} \geq k^n \exp(- O(\sqrt[\ell]{\log n}))$ where $\ell$ is the largest integer such that $2k > 2^\ell$.

math.CO↗

arXiv2010

A new proof of the density Hales-Jewett theorem

The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional grid {1,...,k}^n contains a combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdos and Turan in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg in 1977. The Hales-Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemeredi's theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of {1,2,3}^n of density delta contains a combinatorial line if n is at least a tower of 2's of height O(1/delta^3). Our proof is reasonably simple: indeed, it gives what is arguably the simplest known proof of Szemeredi's theorem.

math.CO↗