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Kevin Ford

Publications and source records attributed to Kevin Ford.

At least 37 records · Page 2Linked to original sources
arXiv2020

Residue classes free of values of Euler's function

We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence of a positive density of odd numbers m, such that for any $s\ge0$ and any even number $a$, the residue class $a\pmod{2^sm}$ contains infinitely many totients.

math.NT↗
arXiv2020

Divisibility of the central binomial coefficient $\binom{2n}{n}$

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. One novelty is a method to capture the effect of large prime factors of integers in general sequences.

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arXiv2020

The distribution of divisors of polynomials

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y 0$ is arbitrarily small.

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arXiv2019

The prime number race and zeros of Dirichlet L-functions off the critical line

Let $π_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally for many triples $(q,a,b)$ that the difference $π_{q,a}(x) - π_{q,b}(x)$ changes sign infinitely often, although there may be a pronounced bias toward one sign (first observed by Chebyshev in 1853). Similar results for the comparison of three or more prime counting functions all require the assumption of ERH (extended Riemann Hypothesis for the Dirichlet L-functions modulo $q$). In this paper we show that the assumption of ERH is, in a sense, necessary. That is, we prove, for any quadruple $(q,a,b,c)$ with $a,b,c$ co-prime to $q$, that the existence of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line imply that one of the six possible orderings of the three functions $π_{q,a}(x), π_{q,b}(x), π_{q,c}(x)$ does not occur at all for large enough $x$.

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arXiv2019

The prime number race and zeros of Dirichlet L-functions off the critical line, II

We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occur for large $x$; see Theorem 5.1.

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arXiv2019

Chebyshev's conjecture and the prime number race

We survey results about prime number races, that is, results about the relative sizes of prime counting functions $π_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.

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arXiv2019

Vinogradov's integral and bounds for the Riemann zeta function

We show for all $1/2 \le σ\le 1$ and $t\ge 3$ that $ζ(σ+it)| \le 76.2 t^{4.45 (1-σ)^{3/2}}$, where $ζ$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by $18.8$. New ingredients include a method of bounding $ζ(s)$ in terms of bounds for Vinogradov's Integral (aka Vinogradov's Mean Value) together with bounds for "incomplete Vinogradov systems", explicit bounds for Vinogradov's integral which strengthen slightly bounds of Wooley (Mathematika 39 (1992), no. 2, 379-399), and explicit bounds for the count of solutions of "incomplete Vinogradov systems", following ideas of Wooley (J. Reine Angew. Math. 488 (1997), 79-140)

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arXiv2019

On two conjectures of Sierpiński concerning the arithmetic functions $σ$ and $ϕ$

Let $σ(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $σ(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpiński from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $ϕ(x)=m$ has exactly $k$ solutions, where $ϕ$ is Euler's function, making progress toward another conjecture of Sierpiński from 1955.

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arXiv2019

Large prime gaps and progressions with few primes

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive this conclusion if there are certain types of exceptional zeros of Dirichlet L-functions.

math.NT↗
arXiv2019

Rough integers with a divisor in a given interval

We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table which are free of prime factors $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.

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arXiv2019

Extremal properties of product sets

We find nearly the optimal size of a set $A\subset [N] := \{1,...,N\}$ so that the product set $AA$ satisfies either (i) $|AA| \sim |A|^2/2$ or (ii) $|AA| \sim |[N][N]|$. This settles problems raised in a recent article of Cilleruelo, Ramana and Ramare.

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arXiv2017

Extreme biases in prime number races with many contestants

We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor classes modulo $q$ so that the corresponding $n$-way prime number race is extremely biased. This improves on the previous range $n \geq φ(q)^ε$, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all $n$-way races being asymptotically unbiased, to biased races existing, occurs when $n = \log^{1+o(1)}q$. The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full $n$-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.

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arXiv2017

Dimensional lower bounds for Falconer type incidence and point configuration theorems

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

math.CA↗
arXiv2017

Permutations contained in transitive subgroups

In the first paper in this series we estimated the probability that a random permutation $π\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $π$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estimate for the proportion of permutations contained in a primitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$.

math.GR↗
arXiv2016

Simultaneous distribution of fractional parts of Riemann zeta zeros

We investigate the simultaneous distribution of the fractional parts of $\{α_1 γ, α_2γ, \cdots, α_nγ\}$, where $n\geq 2$, $α_1$, $α_2$, $\ldots$, $α_n$ are fixed, distinct positive real numbers and $γ$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta-function.

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arXiv2016

Invariable generation of the symmetric group

We say that permutations $π_1,\dots, π_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $π'_1,\dots,π'_r$ of these permutations, $π'_1,\dots,π'_r$ generate $\mathcal{S}_n$. We show that if $π_1,π_2,π_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.

math.GR↗
arXiv2016

Long gaps between primes

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.

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