arXiv subjects
Publications and source records attributed to Kevin Ford.
Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all $x\ge 2y\ge 4$.
Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 - c_r+o(1)}\quad(p\to\infty) $$ for some positive $c_r$ satisfying $c_r\ge e^{-(1+o(1))r} \; (r\to \infty).$
Let $i(n,k)$ be the proportion of permutations $π\in\mathcal{S}_n$ having an invariant set of size $k$. In this note we adapt arguments of the second author to prove that $i(n,k) \asymp k^{-δ} (1+\log k)^{-3/2}$ uniformly for $1\leq k\leq n/2$, where $δ= 1 - \frac{1 + \log \log 2}{\log 2}$. As an application we show that the proportion of $π\in\mathcal{S}_n$ contained in a transitive subgroup not containing $\mathcal{A}_n$ is at least $n^{-δ+o(1)}$ if $n$ is even.
Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$ G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that \[ G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that \[ G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}\] for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.
We show that the counting function of the set of values of the Carmichael $λ$-function is $x/(\log x)^{η+o(1)}$, where $η=1-(1+\log\log 2)/(\log 2)=0.08607...$.
Prior to receiving visual stimuli, spontaneous, correlated activity called retinal waves drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their extended inter-wave intervals and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In contrast with previous, simulation-based models, we are able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how the noise rate and sAHP refractory period contributes to critical wave size variability.
We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-functions. In particular, we show how the rank of an elliptic curve over Q is encoded in the sequences of zeros of other L-functions, not only the one associated to the curve.
We enhance the efficient congruencing method for estimating Vinogradov's integral for moments of order $2s$, with $1\le s\le k^2-1$. In this way, we prove the main conjecture for such even moments when $1\le s\le \tfrac{1}{4}(k+1)^2$, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of $s$, these finding application to the asymptotic formula in Waring's problem.
New version of my 1998 article. The method of proof of the main results follows the original, but there are many simplifications/streamlining of arguments, especially Lemma 3.6 (new Lemma 3.7). Fixed small error in proof of lower bound for V_k(x) (see the paragraph after (5.20)), fixed the statement and proof of Theorem 3. New, precise way to relate sums to volumes (Lemmas 3.1, 3.9) and provided full details in Section 6 of the proofs of Theorems 10-13. Slightly different versions of Theorems 10,11,12,14 (qualitatively the same) with simpler proofs. Sections 7 and 8 combined (new section 7). A few definitions, such as S-normal, changed slightly. Some reorganization of material.
We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.
We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1) random variables and the different v_i need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within O(1).
Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c depends only on the smallest prime not in P. Our proof makes use of results connected with Artin's primitive root conjecture.
For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.
We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $π(x;q,a)<π(x;q,b)$ for a set of real $x$ of asymptotic density 1.
We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.
We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.