arXiv subjects

Kevin Ford

Publications and source records attributed to Kevin Ford.

At least 19 recordsLinked to original sources

arXiv2026

Sign changes of the Liouville function in arithmetic progressions

We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that $λ(m) = -1$ and $λ(n) = + 1$, where $λ$ denotes the Liouville function. Our result is motivated by Heath-Brown's explicit exponent in Linnik's theorem, establishing the existence of primes $p \equiv a \pmod{q}$ with $p \ll q^{5.5}$.

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arXiv2026

Sets of integers satisfying Bateman-Horn statistics

In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n \le x for which f_1(n), ..., f_k(n) are all prime, where (f_1, ..., f_k) is an admissible k-tuple of polynomials in one variable. We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced "f_1(n), ..., f_k(n) are all prime" with "f_1(n), ..., f_k(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful.

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arXiv2026

Poisson Approximation of prime divisors of shifted primes

We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers up to x and of random shifted primes p+a with p < x.

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arXiv2025

Large prime gaps and probabilistic models

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

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arXiv2025

Zero-free regions for the Riemann zeta function

We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the author (Proc. London Math. Soc. 85 (2002), 565-633) for the growth of $ζ(σ+it)$ for $σ$ near 1. Another ingredient is a kind of Jensen formula (Lemma 2.2) relating the growth of a function on vertical lines to a weighted count of its zeros inside a vertical strip. The latter was used recently in the weak subconvexity work of Soundararajan and Thorner (arXiv:1804.03654, Duke Math. J. 168, no. 7 (2019), 1231-1268).

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arXiv2024

Primes with small primitive roots

Let $δ(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le p^{1/4-δ(p)}$ for all $p\in \mathcal{S}$, where $\#\{p\leq x: p\in \mathcal{S}\} \sim π(x)$ as $x\to\infty$.

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arXiv2024

On the theory of prime producing sieves

We develop the foundations of a general framework for producing optimal upper and lower bounds on the sum $\sum_p a_p$ over primes $p$, where $(a_n)_{x/2<n\le x}$ is an arbitrary non-negative sequence satisfying Type I and Type II estimates. Our lower bounds on $\sum_p a_p$ depend on a new sieve method, which is non-iterative and uses all of the Type I and Type II information at once. We also give a complementary general procedure for constructing sequences $(a_n)$ satisfying the Type I and Type II estimates, which in many cases proves that our lower bounds on $\sum_p a_p$ are best possible. A key role in both the sieve method and the construction method is played by the geometry of special subsets of $\mathbb{R}^k$. This allows us to determine precisely the ranges of Type I and Type II estimates for which an asymptotic for $\sum_p a_p$ is guaranteed, that a substantial Type II range is always necessary to guarantee a non-trivial lower bound for $\sum_p a_p$, and to determine the optimal bounds in some naturally occurring families of parameters from the literature. We also demonstrate that the optimal upper and lower bounds for $\sum_p a_p$ exhibit many discontinuities with respect to the Type I and Type II ranges, ruling out the possibility of a particularly simple characterization.

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arXiv2024

A lower bound on the mean value of the Erdős-Hooley Delta function

We give an improved lower bound for the average of the Erdős-Hooley function $Δ(n)$, namely $\sum_{n\le x} Δ(n) \gg_\varepsilon x(\log\log x)^{1+η-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $η= 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{11/4}$ of the second and third authors.

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arXiv2023

Long strings of consecutive composite values of polynomials

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log\log x)^{1/835}$ consecutive integers $n \in [1,x]$ for which $f(n)$ is composite. This improves a result of the first author, Konyagin, Maynard, Pomerance and Tao, which states that there are such strings of length $(\log x)(\log\log x)^{c_f}$, where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of $f$ for some polynomials.

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arXiv2022

Equal sums in random sets and the concentration of divisors

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erdős-Hooley $Δ$-function by $Δ(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $Δ(n) \geq (\log \log n)^{0.35332277\dots}$ for almost all $n$, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $A \subset \mathbb{N}$ by selecting $i$ to lie in $A$ with probability $1/i$. What is the supremum of all exponents $β_k$ such that, almost surely as $D \rightarrow \infty$, some integer is the sum of elements of $A \cap [D^{β_k}, D]$ in $k$ different ways? We characterise $β_k$ as the solution to a certain optimisation problem over measures on the discrete cube $\{0,1\}^k$, and obtain lower bounds for $β_k$ which we believe to be asymptotically sharp.

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arXiv2022

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^δ $ where $δ>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the "trivial" bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^δ$ for sufficiently large $X$, where $δ>0$ depends only on the degree of $f$.

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arXiv2022

Cycle type of random permutations: A toolkit

We prove a number of results, new and old, about the cycle type of a random permutation on S_n. Underlying our analysis is the idea that the number of cycles of size k is roughly Poisson distributed with parameter 1/k. In particular, we establish strong results about the distribution of the number of cycles whose lengths lie in a fixed but arbitrary set I. Our techniques are motivated by the theory of sieves in number theory.

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arXiv2022

Explicit RIP matrices: an update

Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $\varepsilon=3.26\cdot 10^{-7}$, large $k$ and $k^{2-\varepsilon} \le N\le k^{2+\varepsilon}$, we construct $n \times N$ RIP matrices of order $k$ with $k = Ω( n^{1/2+\varepsilon/4} )$.

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arXiv2021

A Hardy-Ramanujan type inequality for shifted primes and sifted sets

We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.

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arXiv2021

Joint Poisson distribution of prime factors in sets

Given disjoint subsets $T_1,\ldots,T_m$ of "not too large" primes up to $x$, we establish that for a random integer $n$ drawn from $[1,x]$, the $m$-dimensional vector enumerating the number of prime factors of $n$ from $T_1,\ldots,T_m$ converges to a vector of $m$ independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when $T_1,\ldots,T_m$ are unrestricted, and apply this to the distribution of the number of prime factors from a set $T$ given that $n$ has $k$ total prime factors.

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arXiv2021

Sets whose differences avoid squares modulo m

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$

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arXiv2020

Solutions of $ϕ(n)=ϕ(n+k)$ and $σ(n)=σ(n+k)$

We show that for some $k\le 3570$ and all $k$ with $442720643463713815200|k$, the equation $ϕ(n)=ϕ(n+k)$ has infinitely many solutions $n$, where $ϕ$ is Euler's totient function. We also show that for a positive proportion of all $k$, the equation $σ(n)=σ(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.

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arXiv2020

Gaps between totients

We study the set D of positive integers d for which the equation $ϕ(a)-ϕ(b)=d$ has infinitely many solution pairs (a,b), where $ϕ$ is Euler's totient function. We show that the minumum of D is at most 154, exhibit a specific A so that every multiple of A is in D, and show that any progression a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies that D equals the set of all positive, even integers.

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