26 Result(s)

Article

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 +...

Kevin Ford in Lithuanian Mathematical Journal (2021)

Article

Dimensional lower bounds for Falconer type incidence theorems

Let 1 ≤ k ≤ d and consider a subset E ⊂ 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, noncongru...

Jonathan DeWitt, Kevin Ford, Eli Goldstein… in Journal d'Analyse Mathématique (2019)

Article

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

Kevin Ford, Adam J. Harper, Youness Lamzouri in Mathematische Annalen (2019)

Article

Extremal Properties of Product Sets

We find the nearly optimal size of a set A ⊂ [N]:= {1,...,N} so that the product set AA satisfies either (i) |AA| ~ |A|²/2 or (ii) |AA| ~ |[N][N]|. This settles problems recently posed in a paper of J. Cilleruelo...

Kevin Ford in Proceedings of the Steklov Institute of Mathematics (2018)

Chapter

Chains of Large Gaps Between Primes

Let p _n denote the n-th prime, and for any k ≥ 1 ...

Kevin Ford, James Maynard, Terence Tao in Irregularities in the Distribution of Prim… (2018)

Article

PREMIS OWL

In this article, we present PREMIS OWL. This is a semantic formalisation of the PREMIS 2.2 data dictionary of the Library of Congress. PREMIS 2.2 are metadata implementation guidelines for digitally archiving ...

Sam Coppens, Ruben Verborgh… in International Journal on Digital Libraries (2015)

Chapter

On the Parity of the Number of Small Divisors of n

For a positive integer j we look at the parity of the number of divisors of n that are at most j, proving that for large j, the count is even for most values of n.

Kevin Ford, Florian Luca, Carl Pomerance, Jeffrey Shallit in Analytic Number Theory (2015)

Chapter

Large Gaps Between Consecutive Prime Numbers Containing Perfect Powers

For any positive integer k, we show that infinitely often, perfect kth powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size ...

Kevin Ford, D. R. Heath-Brown, Sergei Konyagin in Analytic Number Theory (2015)

Article

On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with $${1\leqslant s\leqsla...

Kevin Ford, Trevor D. Wooley in Acta Mathematica (2014)

Article

On common values of φ(n) and σ(m). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the ...

Kevin Ford, Paul Pollack in Acta Mathematica Hungarica (2011)

Article

Prime Chains and Pratt Trees

Prime chains are sequences \(p_{1}, \ldots , p_{k}\) of primes for which

Kevin Ford, Sergei V. Konyagin, Florian Luca in Geometric and Functional Analysis (2010)

Article

On the distribution of imaginary parts of zeros of the Riemann zeta function, II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. ...

Kevin Ford, K. Soundararajan, Alexandru Zaharescu in Mathematische Annalen (2009)

Article

Sharp probability estimates for generalized Smirnov statistics

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line.

Kevin Ford in Monatshefte für Mathematik (2008)

Chapter

From Kolmogorov’s theorem on empirical distribution to number theory

Kevin Ford in Kolmogorov’s Heritage in Mathematics (2007)

Article

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n) ^r = λ(n) ^s , where r

William D. Banks, Kevin Ford, Florian Luca… in Monatshefte für Mathematik (2005)

Article

Maximal Collections of Intersecting Arithmetic Progressions

Let N _t (k) be the maximum number of k-term arithmetic progressions of real numbers, any two of which have t points in common. We determine N ...

Kevin Ford* in Combinatorica (2003)