Kevin Ford's Papers

with selected abstracts




Research Papers

  1. Some infinite series identities . Kevin Ford
    Proc. of the Amer. Math. Soc. 119 (1993), no. 3, 1019-1020.
    Small errata: Must add the hypothesis \( d\ne 1, d\ne -1 \).

  2. The representation on numbers as sums of unlike powers. Kevin Ford
    J. London Math. Soc. (2) 51 (1995), 14-26.
  3. The representation on numbers as sums of unlike powers, II . Kevin Ford
    J. Amer. Math. Soc. 9 (1996), 919-940.
    Addendum and Corrigendum , J. American Math. Soc. 12 (1999), 1213.

  4. New estimates for mean values of Weyl sums . Kevin Ford
    International Mathematical Research Notices 1995, 155-171.
    Abstract: We invent a new method for extracting more efficiently a bound on the \( 2s\)-moment of a degree-k Weyl sum using bounds on Vinogradov's Mean Value. Consequences include large improvements in the known range of validity of the asymptotic formula in Waring's problem.

  5. Sums and products from a finite set of real numbers . Kevin Ford
    The Ramanujan Journal 2, no. 1-2, (1998), 59-66. Reprinted in Analytic and elementary number theory: A tribute to mathematical legend Paul Erdős , (K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum, eds.), Springer, 1998.

  6. The distribution of totients . Kevin Ford
    Note: the PDF version here is the updated 2012 version, with various corrections and simplifications to the original paper.
    The Ramanujan Journal 2, no. 1-2. (1998), 67-151. Reprinted in Analytic and elementary number theory: A tribute to mathematical legend Paul Erdős , (K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum, eds.), Springer, 1998.
    Abstract: A totient is a number in the image of Euler's function \( \phi(n) \). This paper is a comprehensive investigation of the set of totients, solving a number of long-standing open problems. The primary result is a determination of the precise order of growth of \( V(x) \), the number of totients which are at most \( x\). This follows earlier works on the problem going back to 1929 by Pillai, Erdős, Hall, Pomerance and the most recent work of Maier and Pomerance (1988). The formula is rather complex to write down. The proof reveals fine detail about the normal multiplicative structure of a typical totient. We also solve several problems about \( A(m) \), the number of solutions of \( \phi(n)=m \), and \( V_k(x) \), the number of integers \( m\le x \) for which \( A(m)=k \). We show that for every \( k\), either \( V_k(x) \) is identically zero or \( V_k(x) \) has the same order or growth as \( V(x) \), i.e., a positive proportion of totients have multiplicity \( k\) (in fact the latter occurs for all \( k>1 \); see papers No. 7 and 10 below). Moreover, the same results hold if \( \phi \) is replaced by another multiplicative function which has 'similar' behavior at primes, such as the sum of divisors function \( \sigma(n) \). Finally, we study the Carmichael Conjecture (dating to 1907) which states that \( A(m)=1 \) cannot hold for any \( m\). Making use of a large-scale computation, we show that it is true for \( m< 10^{10^{10}} \).

  7. On two conjecture of Sierpinski concerning the arithmetic functions σ and φ , Kevin Ford and Sergei Konyagin.
    In the book Number Theory in Progress (Zakopane, Poland 1997; Kálmán Györy, Henryk Iwaniec, Jerzy Urbanowicz, eds), de Gruyter (1999), 795-803.
    Abstract: Around 1955, W. Sierpiński conjectured that for any positive integer k, there is a number m so that the equation σ(x)=m has exactly k solutions x. Here σ(x) is the sum of the positive divisors of x. In this paper, the conjecture is proved. Sierpiński also conjectured that for every k>1, there is a number m so that the equation φ(x)=m has exactly k solutions x, where φ is Euler's 'totient' function. This conjecture is proved for all even values of k. The proof includes a novel combinatorial device, plus a sieve whose desired output is products of more than one prime (as opposed to prime numbers themselves). See also the companion paper No. 10 below.

  8. Residue classes free of values of Euler's function , Kevin Ford, Sergei Konyagin and Carl Pomerance.
    In the book Number Theory in Progress (Zakopane, Poland 1997; Kálmán Györy, Henryk Iwaniec, Jerzy Urbanowicz, eds), de Gruyter (1999), 805-812.

  9. On integers \( n\) for which \( n \nmid P(n)! \), Kevin Ford
    Unpublished. An earlier version was "published" in the Smarandaceh Notions J. without the author's permission, and subsequently reviewed by Math Reviews.

  10. The number of solutions of φ(x)=m , Kevin Ford
    Annals of Math. 150 (1999), 283-311.
    Abstract: Around 1955, W. Sierpiński conjectured that for any positive integer k>1, there is a number m so that the equation φ(x)=m has exactly k solutions x, where φ is Euler's 'totient' function. This conjecture was proved for all even values of k by the author and Konyagin (see paper No. 7 above). In this paper, the conjecture is proved for every k by a different method. The proof uses induction, using a known integer m with A(m)=k to construct a number m' with A(m')=k+2 (it is similar to a method introduced in paper No. 6 above). One of the main tools is the famous theorem of Chen, used in the following form: for each fixed, even positive integer m, there are at least cx/(log x)2 primes p≤x so that (p-1)/m is prime or the product of two primes.

  11. The Brun-Hooley sieve , Kevin Ford and Heini Halberstam.
    J. Number Theory 81 (2000), 335-350.

  12. Waring's problem with polynomial summands . Kevin Ford
    J. London Math. Soc. (2) 61 (2000), 671-680.

  13. On an irreducibility theorem of A. Schinzel associated with coverings of the integers , Michael Filaseta, Kevin Ford and Sergei Konyagin
    Illinois J. of Math. 44 (2000), 633-643.

  14. Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias , Carter Bays, Kevin Ford, Richard Hudson and Michael Rubinstein.
    J. Number Theory 87 (2001), 54-76.

  15. An explicit sieve bound and small values of σ(φ(m)) . Kevin Ford
    Periodica Math. Hung. 43 (1-2), (2001), 15-29.

  16. Sign changes in πq,a(x) - πq,b(x) , Kevin Ford and Richard Hudson
    Acta Arithmetica 100 (2001), 297-314.

  17. The prime number race and zeros of Dirichlet L-functions off the critical line , Kevin Ford and Sergei Konyagin
    Duke Math. J. 113 (2002), 313-330.
    Abstract: Let πq,a(x) denote the number of primes ≤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 πq,a(x) - πq,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 πq,a(x), πq,b(x), πq,c(x) does not occur at all for large enough x.

  18. Vinogradov's integral and bounds for the Riemann zeta function . Kevin Ford
    Proc. London Math. Soc. 85 (2002), 565-633. List of errata .
    Abstract: We give a new upper bound for the Riemann zeta function near the line ℜs=1. Specifically, we prove the bound |ζ(σ+it)| ≤ A|t|B(1-σ)3/2, with A=76.2 and B=4.45, valid for σ≥ 1/2 and |t|≥ 1. Previously, such a bound was known with B=18.8 (the value of B is far more important in applications). Applications include improvements to error terms in the divisor problems, and numerical improvements to zero-free regions for ζ(s); see paper No. 19 below. The main new ingredient is the utilization of multidimensional exponential sums over smooth numbers (numbers without large prime factors).

  19. Zero-free regions for the Riemann zeta function . Kevin Ford
    In the book Number Theory for the Millenium (Urbana, IL, 2000) (M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Phillip, eds), vol. II (2002), 25-56.

  20. Maximal collections of intersecting arithmetic progressions . Kevin Ford
    Combinatorica 23 (2) (2003), 263-281.

  21. The jumping champions of the Farey series, Christian Cobeli, Kevin Ford and Alexandru Zaharescu
    Acta Arithmetica 110 (2003), no. 4, 259-274

  22. The prime number race and zeros of Dirichlet L-functions off the critical line, II. , Kevin Ford and Sergei Konyagin
    Proceedings of the session in analytic number theory and Diophantine equations (Bonn, January-June 2002), Bonner Mathematische Schiften, Nr. 360 (D. R. Heath-Brown, B. Z. Moroz, editors), Bonn, Germany, 2003, 40 pages.

  23. A strong form of a problem of R. L. Graham . Kevin Ford
    Canadian Math. Bulletin 47 (3), (2004), 358-368.

  24. On Bombieri's asymptotic sieve . Kevin Ford
    Trans. Amer. Math. Soc. 357 (2005), 1663-1674.

  25. Cycling via common divisors; Solution of Problem 11019(b), Kevin Ford and Sergei Konyagin
    Amer. Math. Monthly 111 (2005), 277-278.
    Notes: This file corrects a few typos in the published version.

  26. On the distribution of imaginary parts of zeros of the Riemann zeta function , Kevin Ford and Alexandru Zaharescu
    J. reine angew. Math. 579 (2005), 145-158.

  27. On the maximal difference between an element and its inverse in residue rings . Kevin Ford, Mizan R. Khan, Igor E. Shparlinski and Christian L. Yankov
    Proc. Amer. Math. Soc. 133 (2005), 3463-3468.

  28. Values of the Euler function in various sequences William D. Banks, Kevin Ford, Florian Luca, Francesco Pappalardi and Igor E. Shparlinski,
    Monatshefte Math. 146 (2005), 1-19.

  29. Sieving by large integers, and covering systems of congruences , Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance and Gang Yu
    J. Amer. Math. Soc. , 20 (2007), 495-517.
    Abstract: A covering system is a finite set of congruence classes, say \( a_j\mod m_j, 1\le j\le k \), whose union is all the integers. Of particular interest are those covering systems with distinct moduli \( m_j \). The concept was invented by Erdős, who has a number of famous problems about them. In this paper, we prove a conjecture of Erdős and Ron Graham, that for any \(K>1\), if \(N\) is sufficiently large then there does not exists a covering system with distinct moduli from the interval \( (N,KN \). We also prove a conjecture of Erdős and John Selfridge, that for any \(K>1\), if \(N\) is sufficiently large then there does not exists a covering system with distinct moduli larger than \(N\) whose sum of reciprocals is less than \(K\). In fact, in each of these theorems, the quantity \(K\) may grow as a function of \(N\) (almost linearly), and each modulus may be be repeated \(s\) times, with \(s\) slowly growing as a function of \(N\). These are accomplished using new methods of bounding the density of the union of congruence classes, when the moduli are not pairwise coprime.

  30. The distribution of integers with at least two divisors in a short interval , Kevin Ford and Gérald Tenenbaum
    Quart. J. Math. Oxford 58 (2007), 187-201.

  31. Generalized Smirnov statistics and the distribution of prime factors . Kevin Ford
    Funct. Approx. Comm. Math. 37.1 (2007), 119-129.

  32. Localized large sums of random variables , Kevin Ford and Gérald Tenenbaum
    Stat. Prob. Letters 78 (2008), 84-89.

  33. Sharp Probability estimates for generalized Smirnov statistics . Kevin Ford
    Monatshefte Math. 153 (2008), 205-216.

  34. The distribution of integers with a divisor in a given interval . Kevin Ford
    Annals of Math. (2) 168 (2008), 367-433.
    Abstract: This paper solves several central problems of Erdős about the distribution of divisors of integers. Let \(H(x,y,z)\) denote the number of integers \(n\le x\) that have a divisor between \(y\) and \(z\). The main theorem in this paper is a determination of the exact order of growth of \(H(x,y,z)\) for all \(x,y,z\). (that is, bounding \(H(x,y,z)\) between two constant multiples of an explicit smooth function of \(x,y,z\) ). This caps off a program begun by Besicovich in the 1930s, and earlier estimates of Erdős, Tenenbaum and others. In particular, in the important special case \( z=2y\), \( H(x,y,2y) \) has order \[ \frac{x}{(\log y)^{\mathcal{E}}(\log\log y)^{3/2}}, \] uniformly for \( 3\le y\le x^{1/2} \), where \(\mathcal{E} = 1-(1+\log\log 2)/\log 2=0.086071332...\)

    One application is the solution of the Erdős "multiplication table problem" (posed in 1955): let \(A(N)\) be the number of distinct entries in an \(N\) by \(N\) multiplication table. Then \(A(N)\) lies between two constant multiples of \[ \frac{N^2}{(\log N)^{\mathcal{E}} (\log\log N)^{3/2}} \] We also determine the order of \(H_r(x,y,z)\), the number of integers \(n\le x\) that have exactly \( r\) divisors between \(y\) and \(z\), in a wide range of the parameters, including the central case \( z=2y \). In this case, we show that for every \( r \ge 1 \), \( H_r(x,y,2y) \) has the same order as \( H(x,y,2y) \), disproving a 1960 conjecture of Erdős. A key tool in the proofs is a new avoidance bound for random walks.

    The new bounds for random walks was generalized and strengthened in sequel papers devoted to random walk avoidance theory (nos. 31, 33, 42 in this list).

    The techniques of this paper were also used to provide tight bounds on the probability that a random permutation on n letters has a fixed set of a given size: Permutations fixing a k-set (2016).

  35. Integers with a divisor in (y,2y] . Kevin Ford
    Anatomy of Integers (Jean-Marie De Koninck, Andrew Granville, and Florian Luca, eds), CRM Proc. and Lect. Notes 46 , Amer. Math. Soc., Providence, RI, 2008, 65-80. Updated, simplified version (2012)
    Abstract: This is an alternative and simplified proof of a central case of the main theorem of paper No. 34 above, namely the determination of the order of growth of H(x,y,2y), the number of integers n≤x that have a divisor between y and 2y. This paper will be periodically updated, as I or others come up with simplifications.

  36. Generalized Euler constants , Kevin Ford and Harold G. Diamond
    Math. Proc. Cambridge Phil. Soc. 145 (2008), 27-41.

  37. Divisors of the Euler and Carmichael functions, Kevin Ford and Yong Hu
    Acta Arithmetica 133 (2008), 199-208.

  38. On curves over finite fields with Jacobians of small exponent, Kevin Ford and Igor E. Shparlinski
    Int. J. Number Theory 4 (2008), 819-826.

  39. On the distribution of imaginary parts of zeros of the Riemann zeta function, II , Kevin Ford, K. Soundararajan and Alexandru Zaharescu
    Math. Annalen 343 (2009), 487-505.

  40. Diophantine approximation with arithmetic functions , Emre Alkan, Kevin Ford and Alexandru Zaharescu
    Trans. Amer. Math. Soc. 361 (2009), 2263-2275.

  41. On the largest prime factor of the Mersenne numbers , Kevin Ford, Florian Luca and Igor E. Shparlinski
    Bull. Austr. Math. Soc. 79 (2009), 455-463.

  42. Sharp probability estimates for random walks with barriers . Kevin Ford
    Prob. Th. Rel. Fields 145 (2009), 269-283.

  43. Diophantine approximation with arithmetic functions, II , Emre Alkan, Kevin Ford and Alexandru Zaharescu
    Bull. London Math. Soc. 41 (2009), 676-682.

  44. Common values of the arithmetic functions \( \phi \) and \( \sigma \) , Kevin Ford, Florian Luca and Carl Pomerance
    Bull. London Math. Soc. 42 (2010), 478-488.

    Abstract: Do the ranges of Euler's totient function \( \phi \) and the sum-of-divisors function \( \sigma \) have infinite intersection? In 1958, Erdős conjectured that they do. This follows trivially from either of two other famous conjectures, the twin prime conjecture (if \( p\) and \( p+2 \) are both prime, then \( \phi(p+2)=p+1=\sigma(p) \) and the conjecture that there are infinitely many Mersenne primes (if \( 2^p-1 \) is prime, then \( \phi(2^{p+1}) = 2^p = \sigma(2^p-1) \) . Both of these conjectures are out of reach. Numerical evidence indicates that intersections are plentiful, however, in fact below 107 about half of all numbers in the range of \( \phi \) are also in the range of \( \sigma \). In this paper, we prove Erdős conjecture unconditionally. One key tool is an upper bound for the number of constrained prime chains from paper No. 47 (below).

  45. Geometric Properties of Points on Modular Hyperbolas , Kevin Ford, Mizan Khan and Igor Shparlinski
    Proc. Amer. Math. Soc. 138 (2010), 4177-4185.

  46. On the divisibility of Fermat quotients , Jean Bourgain, Kevin Ford, Sergei V. Konyagin and Igor E. Shparlinski
    Michigan Math. J. 59 (2010), 313-328.

  47. Prime chains and Pratt trees , Kevin Ford, Sergei V. Konyagin and Florian Luca
    Geom. Funct. Anal. 20 (2010), 1231-1258.
    Abstract: A prime chain is a sequence \( p_1, p_2, \ldots, p_k \) of primes such that \( p_j | (p_{j+1}-1) \) for each \( j\le k-1 \). We introduce a number of new methods for counting prime chains with various constraints, primarily with the length \( k \) unconstrained. For example, the number of prime chains starting at a given prime \( p \) and ending at a prime less than \( xp \) is, for any \( \varepsilon>0 \), at most \( C(\varepsilon) x^{1+\varepsilon} \) for some constant \( C(\varepsilon) \) ; this bound has an application to common values of the functions \( \phi(n) \) and \( \sigma(n) \) (See above). We also study the distribution of Pratt trees , the tree structure formed by all of the prime chains ending at a given prime \( p \) (and related to the Pratt certificate of primality). We prove the first nontrivial bounds on the height of the Pratt tree, valid for almost all primes \( p\), and also develop a sophisticated stochastic model of the Pratt trees based on branching random walks. This model leads us to conjecture various behaviors of the Pratt trees, in particular that the heights have a tight probability distribution with mean close to \( e \log \log p \).

  48. Chebyshev's bias for products of two primes , Kevin Ford and Jason Sneed
    Experim. Math. 19 (2010), 385-398.

  49. The number of solutions of λ(x)=n , Kevin Ford and Florian Luca
    INTEGERS (Proceedings of the INTEGERS conference honoring Melvyn Nathanson and Carl Pomerance) 11A (2011), Article 9.

  50. Explicit constructions of RIP matrices and related problems , Jean Bourgain, Stephen J. Dilworth, Kevin Ford, Sergei Konyagin and Denka Kutzarova
    Duke Math. J. 159 (2011), 145-185.
    Abstract: We break a natural barrier in the explicit construction of matrices useful in compressed sensing. Let \( \Phi \) be an \( N \times n \) matrix. Given parameters \( k \) and \( \delta \), if \[ (1-\delta) \| \mathbf{x} \| \le \| \Phi \mathbf{x} \| \le (1+\delta) \| \mathbf{x} \|\] for any vector \( \mathbf{x}\in \mathbb{C}^N \) with at most \( k\) nonzero coefficients, the matrix \( \Phi \) is said to satisfy the Restricted Isometry Property (RIP) of order \( k\) and constant \( \delta \). If the entries of \( \Phi \) are chosen randomly, then with high probability, \( \Phi \) will satisfy the RIP property if order \( k > cn/\log N. \) All known explicit constructions do much worse, only producing the RIP property of order \( k = O(n^{1/2}) \) due to the use of coherence (requiring the columns to have pairwise small inner products). In this paper, we construct a new type of matrix that provably satisfies the RIP property with larger order. Specifically, for some specific \( \varepsilon>0 \), large \( N \) and any \( n \) satisfying \( N^{1-\varepsilon} \le n\le N \), we construct RIP matrices of order \( k = n^{1/2+\varepsilon} \) and constant \( \delta= n^{-\varepsilon} \). Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums over sets possessing special additive structure.

    In the same paper, we also give a construction of sets of \( n\) complex numbers whose \( k\)-th moments are uniformly small for \( 1\le k\le N \) (Turán's power sum problem), which improves upon known explicit constructions when \( N\) is very large compared to \( n\).

  51. Breaking the k2 barrier for explicit RIP matrices , Jean Bourgain, Stephen J. Dilworth, Kevin Ford, Sergei Konyagin and Denka Kutzarova
    Proceedings of the 43rd Symposium on the Theory of Computing (STOC '11) (2011), 637-644.
    Abstract: This is a summary of the main ideas from the previous paper No. 50. In addition, some new ideas are inserted in a key step which allows one to greatly improve the value of the constant ε appearing in the main theorem (it is still rather tiny, as it depends on a bound in additive combinatorics which is expected to be far from optimal).

  52. On common values of \( \phi(n) \) and \( \sigma(m) \), I . Kevin Ford and Paul Pollack
    Acta Math. Hungarica 133 (2011), 251-271.

  53. A problem of Ramanujan, Erdős and Katai on the iterated divisor function , Yvonne Buttkewicz, Christian Elsholtz, Kevin Ford and Jan-Christoff Schlage-Puchta
    International Mathematical Research Notices 2012 , 4051-4061.

  54. On common values of \( \phi(n) \) and \( \sigma(m) \), II . Kevin Ford and Paul Pollack
    Algebra and Number Theory 6 (2012), no. 8, 1669-1696.
    Abstract: Computational evidence suggests that a positive proportion of the integers in the range of Euler's function \( \phi(n) \) are also in the range of the sum-of-divisors function \( \sigma(m) \). In fact, up to \( 10^8 \), roughly half of all integers in the range of \( \phi \) are also in the range of \( \sigma \) and vice-versa. This is misleading, however. In this paper, we prove that the intersection of the two ranges has relative density zero inside the range of \( \phi \) and inside the range of \( \sigma \). The proof requires the full structure theory of typical \( \phi- \) and \(\sigma\)-values from paper No. 6 above.

  55. On groups with perfect order subsets , Kevin Ford, Sergei Konyagin and Florian Luca.
    Moscow J. Comb. Number Th. 2 (2012), 297-312.

  56. Poisson-Dirichlet branching random walks . Louigi Addario-Berry and Kevin Ford
    Ann. Appl. Prob. 23 , (1) (2013), 283-307.

  57. On square values of the product of the Euler totient and sum of divisors functions . Kevin Broughan, Kevin Ford and Florian Luca
    Colloq. Math. 130 (2013), 127-137.

  58. On non-intersecting arithmetic progressions , Régis de la Bretèche, Kevin Ford and Joseph Vandehey
    Acta Arithmetica 157 (2013), 381-392.

  59. The prime number race and zeros of L-functions off the critical line. III. Kevin Ford, Sergei Konyagin and Youness Lamzouri,
    Quart. J. Math. Oxford 64 (2013), 1091-1098.

  60. Values of the Euler φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields , Kevin Ford, Florian Luca and Pieter Moree
    Math. Comp. 83 (2014), no. 287, 1447-1476.

  61. Sieving very thin sets of primes, and Pratt trees with missing primes . Kevin Ford
    International Mathematical Research Notices 2014 (2014), 2955-2971.

  62. On Vinogradov's mean value theorem: strongly diagonal behaviour via efficient congruencing , Kevin Ford and Trevor D. Wooley
    Acta Mathematica 213 (2014), 199-236.
    Abstract: I. M. Vinogradov introduced in the 1930s the quantity \( J_{s,k}(N) \), which counts the integer solutions of the system of equations \( \sum_{i=1}^s (x_i^j-y_i^j)=0 \quad (1\le j\le k) \), where \( 1\le x_i,y_i \le N \) for all \( i\) Estimates for the number of solutions are particularly useful in Diophantine problems such as Waring's problem, and have also led to the best known zero-free region for the Riemann zeta function (see above). The Main Conjecture for Vinogradov's system states that \[ J_{s,k}(N) \ll_\epsilon N^\epsilon (N^s + N^{2s-k(k+1)/2}). \] In 2011, Wooley introduced a new, powerful method known as efficient congruencing which led to a proof of this conjecture for \( s \ge k^2 \). In the "small s" range, that is \( s\le \frac12 k(k+1) \), it is conjectured that the "diagonal" solutions (solutions with \(x_1,\ldots,x_s \) a permutation of \(y_1,\ldots,y_s\) ) dominate and consequently that \( J_{s,k}(X) \ll_\epsilon X^{s+\epsilon} \). Until recently this was know only in the trivial case \( s \le k\) and for \( s=k+1 \) . In this paper, we develop a variant of the efficient congruencing method to prove this conjecture in the much wider range \(1\le s\le \frac14 (k+1)^2\).

  63. The image of Carmichael's \(\lambda\)-function , Kevin Ford, Florian Luca and Carl Pomerance
    Algebra and Number Theory 8 (2014), 2009-2025.
    Notes: A few small typos have been corrected in the version here.
    Abstract: We show that the counting function of the range of Carmichael's \(\lambda\)-function is \( x (\log x)^{-\mathcal{E}+o(1)}\), where \( \mathcal{E} = 1 - \frac{1+\log\log 2}{\log 2} = 0.086071332\ldots\) is the same constant appearing in the multiplication table problem (see paper #34 above).

  64. Unnormalized differences between zeros of L-functions , Kevin Ford and Alexandru Zaharescu
    Compositio Math. 151 (2015), 230-252.

  65. Large gaps between consecutive prime numbers containing perfect powers , Kevin Ford, D. R. Heath-Brown and Sergei Konyagin
    in the book Analytic Number Theory, in honor of Helmut Maier's 60th birthday, C. Pomerance and M.Th. Rassias (eds.), Springer-Verlag, 2015, pp. 83-92.

  66. On the parity of the number of small divisors of n , Kevin Ford, Florian Luca, Carl Pomerance and Jeffrey Shallit
    in the book Analytic Number Theory, in honor of Helmut Maier's 60th birthday, C. Pomerance and M.Th. Rassias (eds.), Springer-Verlag, 2015, pp. 93-100.

  67. Large gaps between consecutive prime numbers . Kevin Ford, Ben Green, Sergei Konyagin and Terence Tao
    Annals of Math. 183 (2016), 935-974.
    Abstract: Let \( G(x) \) denote the largest gap between consecutive prime numbers which are less than \(x \). A famous lower bound of Rankin from 1938 (itself an improvement upon earlier results of Westzynthius and Erdős) states that \[ G(x) \ge c \frac{\log x \log\log x \log\log\log\log x}{(\log\log\log x)^2} \] for some positive constant \(c \). In the next 75 years, only the constant \( c \) was improved, the best result being that of Pintz from 1997 that \( c=2e^{\gamma}-o(1) \). Because the above limit was the natural limitation of a certain circle of ideas, Erdős offered a prize of $10,000 to prove that the constant \(c\) could be taken arbitrarily large (that is, for every \( c\) the above bound holds if \( x\) is sufficiently large in terms of \( c\) ). In this paper, we solve this conjecture. The proof uses a novel probabilistic construction, together with ideas stemming from the recent breakthroughs of Green and Tao for counting solutions of linear equations over primes.
    Media publicity for this paper:

  68. Permutations fixing a k-set , Sean Eberhard, Kevin Ford and Ben Green
    IMRN 2016 , no. 21, 6713-6731.
    Abstract: Adapting methods from my paper The distribution of integers with a divisor in a given interval, we determine the exact order of the number of permutations in \(S_n\) which have a fixed set of size \(k\), uniformly in \(n,k\). We say that \(\sigma\in S_n\) has a fixed set of size \(k\) if there is a subset of \( \{1,\ldots,n\}\) which is left invariant by \(\sigma\). Equivalently, there is a subset of the cycles in \(\sigma\) with sum of lengths equal to \(k\).

  69. Permutations contained in transitive subgroups , Sean Eberhard, Kevin Ford and Dimitrios Koukoulopoulos
    Discrete Analysis 2016: 12 , 34 pages.

  70. Invariable generation of the symmetric group , Sean Eberhard, Kevin Ford and Ben Green
    Duke Math. Journal 166 (2017), no. 8, 1573-1590.
    Abstract: Permutations \(\sigma_1,\ldots,\sigma_r\) are said to invariably generate the symmetric group \(S_n\) if any conjugates \(\sigma_1',\ldots,\sigma_r' \) generate \(S_n\). Pemantle, Peres and Rivin showed that with probability at least \( \delta \), a positive absolute constant, four random permutations of \(S_n\) invariably generate \(S_n\). We show that three random permutations do not suffice, that is, the probability that random \(\sigma_1,\sigma_2,\sigma_3\) in \(S_n\) generate \(S_n\) tends to zero as \(n\to\infty\).

  71. Integers divisible by a large shifted prime . Kevin Ford.
    Acta Arithmetica 178 (2017), no. 2, 163-180.

  72. On the smallest simultaneous power nonresidue modulo a prime . Kevin Ford, Moubariz Z. Garaev and Sergei V. Konyagin,
    Forum Mathematicum 29 (2017), no. 2, 347-355.

  73. Simultaneous distributions of fractional parts of Riemann zeta zeros , Kevin Ford, Xianchang Meng and Alexandru Zaharescu
    Bulletin of the London Math. Soc. 49 , no. 1, (2017), 1-9.

  74. Long gaps between consecutive prime numbers , Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and and Terence Tao,
    J. Amer. Math. Soc., 31 (2018), no. 1, 65-105.
    Abstract: In this paper, we make a quantitative improvement to the lower bound for \( G(x) \), the largest gap between consecutive primes less than \(x \), see paper #67 above. We prove that for some constant \( c\) and large \( x\), \[ G(x) \ge c \frac{\log x \log\log x \log\log\log\log x}{\log\log\log x} \] improving the previous bound by a factor of \( \log\log\log x \). There are three main ingredients in the proof. First, we utilize the probabilistic constructions from the earlier paper No. 67 above. Next, we incorporate a uniform version of the multidimensional prime detecting sieve method of Maynard which was earlier used to show that there are many primes. thirdly, we prove a generalization of a hypergraph covering theorem of Pippenger and Spencer, which allows for an essentially optimal means of translating such estimates into a result on large gaps between primes.

  75. Chains of large gaps between primes , Kevin Ford, James Maynard and Terence Tao
    in the book "Irregularities in the distribution of prime numbers", (J. Pintz, M. Th. Rassias, eds.), Springer, 2018, p. 1-21.

  76. Dimensional lower bounds for Falconer type incidence and point configuration theorems . Jonathan DeWitt, Kevin Ford, Eli Goldstein, Steven J. Miller, Gwyneth Moreland, Eyvindur A. Palsson, and Steven Senger. J. d'Analyse Math. 139 (2019), 143-154.

  77. Extreme biases in prime number races with many contestants. Kevin Ford, Adam J. Harper and Youness Lamzouri. Math. Annalen 374 (1) (2019), 517-551.
    Abstract: Under the assumptions of GRH and LI (linear independence of imaginary parts of Dirichlet L-functions), we show that for any large modulus \(q\) and \( \log q \le r \le q^{1/50} \), there are residues \( a_1,\ldots,a_r \bmod q\) so that some ordering of the prime counting functions \( \pi(x;q,a_i) \) occurs with density at most \( (1/r!)\exp\{ - c r /\log q \} \), and there is another set of residues \( b_1,\ldots,b_r \bmod q \) such that some ordering of the prime counting functions \( \pi(x;q,b_i) \) occurs with density at least \( (1/r!)\exp\{ c r /\log q \} \). Here \( c\) is a positive, absolute constant. As there are \( r! \) orderings of these sets of prime counting functions, this shows an extreme bias towards or away from certain orderings.

  78. Extremal properties of product sets. Kevin Ford. Proceedings of the Steklov Mathematics Institute, Trudy Matematicheskogo Instituta imeni V.A. Steklova 303 , no. 1 (2018), 220-226. special issue dedicated to the 60th birthday of Sergei V. Konyagin. to appear.

  79. Long gaps in sieved sets . Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance and Terence Tao. J. European Math. Soc. , to appear.
    Abstract: We create a new method for detecting large gaps in sequences of integers where an average of one residue class per prime has been omitted (sifted out). As a consequence, we deduce that for any irreducible polynomial \( f: \mathbb{Z}\to\mathbb{Z} \) with positive leading coefficient, if \(x\) is large then there are strings of \( (\log x)(\log\log x)^c \) consecutive values of \( n \in [1,x] \) for which \( f(n) \) is composite. Here \(c\) is a positive quantity depending only on the degree of \(f\). Previously, only gaps of size \( \gg \log x \) were known to exist, following a simple argument.

  80. Rough integers with a divisor in a given interval. 17-Sept-2019 revision. Kevin Ford. 17 pages. J. Australian Math. Soc. , to appear.

  81. Large prime gaps and progressions with few primes. Kevin Ford. Rivista di Matematica della Universita di Parma , to appear.

  82. Equal sums in random sets and the concentration of divisors. Kevin Ford, Ben Green and Dimitris Koukoulopoulos. 89 pages. (Aug. 1, 2019)
    Abstract: Let \(A\subset \mathbb{N}\) be a random set, where \(n\) is included in \(A\) with probability \(1/n\). We study the problem of determining for which intervals \( [X,Y] \), the subset \( A \cap [X,Y] \) will have \(k\) equal subset sums. We apply our theorems to prove new lower bounds on the maximum concentration of divisors of integers, of permutations and of polynomials over a finite field. In particular, we disprove a conjecture of Maier and Tenenbaum from 2009 about the normal order of the Erdős-Hooley function \( \Delta(n)\), which is the maximum number of divisors of \(n\) lying in any interval \( (u,eu] \) of logarithmic length 1, by showing that \( \Delta(n) \ge (\log\log n)^{B+o(1)} \) for almost all \(n\), where \(B\) is a specific constant which is about 0.35332. Our analysis of the subset sum problem involves linear algebra on the \(k\)-dimensional unit cube \( \{0,1\}^k \) and an optimisation problem over measures on \( \{0,1\}^k \). We believe that our results are asymptotically sharp, that is, that \( \Delta(n)=(\log\log n)^{B+o(1)} \) for almost all \(n\).

  83. Large prime gaps and probabilistic models . Bill Banks, Kevin Ford and Terence Tao, 38 pages.
    Abstract: We introduce a new 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. This differs fundamentally from the well-known model of Cramer and the refined model of Granville. 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. In particular, if the true order of the interval sieve problem is smaller than expected, we predict that the largest gap between primes below \(x\) grows faster than \( c\log^2 x \) for any positive \(c\). The same methods also allow us to rigorously relate the validity of a uniform version of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

  84. Divisibility of the central binomial coefficient \( \binom{2n}{n} \). Kevin Ford and Sergei Konyagin. 24 pages. Trans. Amer. Math. Soc., to appear.
    Abstract: We show that for any positive integer \(\ell\), the set of integers \(n\) satisfying \( n^\ell | \binom{2n}{n} \) has a positive density. We also determine asymptotically the counting function of \(n\) such that \( (n,\binom{2n}{n})=1 \).

  85. The distribution of divisors of polynomials . Kevin Ford and Guouyou Qian. Mathematika 66 (2020), 395-415.

  86. Solutions of \( \phi(n)=\phi(n+k) \) and \( \sigma(n)=\sigma(n+k) \) , Kevin Ford, IMRN, to appear.

  87. Gaps between totients , Kevin Ford and Sergei Konyagin. preprint.

  88. Sets whose differences avoid squares modulo m , Kevin Ford and Mikhail Gabdullin, preprint.

Expository/Survey/Announcement Papers

  1. The distribution of totients . Kevin Ford
    ERA Amer. Math. Soc. , 4 (1998), 27-34.
    Abstract: This is an announcement and extended abstract of papers No. 6, 7 and 10 above.

  2. Chebyshev's conjecture and the prime number race , Kevin Ford and Sergei Konyagin
    Modern Problems of Number Theory and its Applications; Topical Problems Part II (Tula, Russia, 2001). (2002), 67-91.

  3. Recent progress on the estimation of Weyl sums Kevin Ford
    Modern Problems of Number Theory and its Applications; Topical Problems Part II (Tula, Russia, 2001). (2002), 48-66.

  4. From Kolmogorov's theorem on empirical distribution to number theory . Kevin Ford
    in the book "Kolmogorov's legacy in mathematics", eds. E. Charpentier et al., Editions Belin (Paris), 2004, 111-120. The printed version is in French. PDF file of English original (8 pages). See below for an updated version (appeared in the English version of the book).

  5. From Kolmogorov's theorem on empirical distribution to number theory . Kevin Ford
    Appeared in the book Kolmogorov's heritage in mathematics (English edition) , Editions Belin/Springer, 2007, 97-108. Updated version (with sketch of a different proof of the main result).
    Abstract: This paper describes how new bounds for constrained random walks (related to Kolmogorov-Smirnov statistics) are used to study the distribution of divisors of integers in paper No. 34.

Unpublished work

  1. Simple proof of Gallagher's singular series sum estimate , unpublished 1-page note, 13-Oct-2007, arXiv:1108.3861