arXiv subjects
Publications and source records attributed to Nathan McNew.
We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable. We also present some related unsolved problems.
We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering numbers possesses a natural density $d(\mathcal{C})$ and prove that $0.103230 < d(\mathcal{C}) < 0.103398.$ Our approach adapts methods developed by Behrend and Del\'eglise for bounding the density of abundant numbers, by introducing a function $c(n)$ that measures how close an integer $n$ is to being a covering number with the property that $c(n) \leq h(n) = \sigma(n)/n$. However, computing $d(\mathcal{C})$ to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for $d(\mathcal{A})$, the density of abundant numbers, namely $0.247619608 < d(\mathcal{A}) < 0.247619658$. We also show the count of primitive covering numbers up to $x$ is $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + \epsilon\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers.
In 1977 Diaconis and Graham proved two inequalities relating different measures of disarray in permutations, and asked for a characterization of those permutations for which equality holds in one of these inequalities. Such a characterization was first given in 2013. Recently, another characterization was given by Woo, using a topological link in $\mathbb R^3$ that can be associated to the cycle diagram of a permutation. We show that Woo's characterization extends much further: for any permutation, the discrepancy in Diaconis and Graham's inequality is directly related to the Euler characteristic of the associated link. This connection provides a new proof of the original result of Diaconis and Graham. We also characterize permutations with a fixed discrepancy in terms of their associated links and find that the stabilized-interval-free permutations are precisely those whose associated links are nonsplit.
Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields and polynomial rings over finite fields. We study the analogous problem in the Hurwitz quaternion order to see how non-commutativity affects the problem. We compute an exact formula for the density of a 3-term geometric-progression-free set of Hurwitz quaternions arising from a greedy algorithm and derive upper and lower bounds for the supremum of upper densities of 3-term geometric-progression-free sets of Hurwitz quaternions.
Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all $k \ge 2$. We also develop polynomial versions of the classical techniques used to study gaps between $k$-free integers in $\mathbb Z$. We apply these techniques to obtain analogues in $\mathbb F_q[x]$ of some classical theorems on the distribution of $k$-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.
We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree integer. The constant 11 can be improved further, if $x$ is assumed to be larger than a (very) large constant.
Let $P^{\left(\frac 12\right)}(n)$ denote the middle prime factor of $n$ (taking into account multiplicity). More generally, one can consider, for any $α\in (0,1)$, the $α$-positioned prime factor of $n$, $P^{(α)}(n)$. It has previously been shown that $\log \log P^{(α)}(n)$ has normal order $α\log \log x$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\leq x$ for which $P^{(α)}(n)=p$, for primes $p$ in a wide range up to $x$. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac 1x \sum_{1<n \le x} \log P^{\left(\frac 12 \right)}(n) \sim A(\log x)^{φ-1}$, where $φ$ is the golden ratio, and $A$ is an explicit constant. Along the way, we obtain an extension of Lichtman's recent work on the ``dissected'' Mertens' theorem sums $\sum_{\substack{P^+(n) \le y \\ Ω(n)=k}} \frac{1}{n}$ for large values of $k$.
Let $S_{\rm div}(n)$ denote the set of permutations $π$ of $n$ such that for each $1\leq j \leq n$ either $j \mid π(j)$ or $π(j) \mid j$. These permutations can also be viewed as vertex-disjoint directed cycle covers of the divisor graph $\mathcal{D}_{[1,n]}$ on vertices $v_1, \ldots, v_n$ with an edge between $v_i$ and $v_j$ if $i\mid j$ or $j \mid i$. We improve on recent results of Pomerance by showing $c_d = \lim_{n \to \infty }\left(\# S_{\rm div}(n)\right)^{1/n}$ exists and that $2.069<c_d<2.694$. We also obtain similar results for the set $S_{\rm lcm}(n)$ of permutations where ${\rm lcm}(j,π(j))\leq n$ for all $j$. The results rely on a graph theoretic result bounding the number of vertex-disjoint directed cycle covers, which may be of independent interest.
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = α^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $α$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
Noting that cycle diagrams of permutations visually resemble grid diagrams used to depict knots and links in topology, we consider the knot (or link) obtained from the cycle diagram of a permutation. We show that the permutations which correspond in this way to an unknot are enumerated by the Schröder numbers, and also enumerate the permutations corresponding to an unlink. The proof uses Bennequin's inequality.
Erdős proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that $\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$, where $\mathcal{P}_j$ is the set of integers with $j$ prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field $\mathbb{F}_q[x]$, investigating the sum $\mathcal{F}(A) := \sum_{f \in A} \frac{1}{\text{deg} f \cdot q^{\text{deg} f}}$. We establish a uniform bound for $\mathcal{F}(A)$ over all primitive sets of polynomials $A \subset \mathbb{F}_q[x]$ and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for $q = 2, 3$, and $4$, but we find computational evidence that it holds for $q > 4$.
A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we generalize a result of Besicovitch to show that there exist primitive sets in $\mathbb{F}_q[x]$ with upper density arbitrarily close to $\frac{q - 1}{q}$. Then, for a primitive set $A$, we consider the sum $\sum_{a \in A} \frac{1}{q^{°a}°a}$, the natural analogue in this setting of a sum considered by Erdős for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets $A$. We end with a generalization of work of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the $k$-th irreducible polynomial.
A partition $α$ is said to contain another partition (or pattern) $μ$ if the Ferrers board for $μ$ is attainable from $α$ under removal of rows and columns. We say $α$ avoids $μ$ if it does not contain $μ$. In this paper we count the number of partitions of $n$ avoiding a fixed pattern $μ$, in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $μ$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic $p$-groups. We further obtain asymptotics for the number of partitions of $n$ avoiding a pattern $μ$. Using these asymptotics we conclude that the generating function for $μ$ is not algebraic whenever $μ$ is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.
A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage are. We explore sufficient conditions on infinite sets of positive integers such that there are either no sum-dominant subsets, at most finitely many sum-dominant subsets, or infinitely many sum-dominant subsets. In particular, we prove no subset of the Fibonacci numbers is a sum-dominant set, establish conditions such that solutions to a recurrence relation have only finitely many sum-dominant subsets, and show there are infinitely many sum-dominant subsets of the primes.
We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_1(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_1(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_1(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the other hand however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_1(n)$ tends to infinity on the integers outside this set. Finally we consider the asymptotic behaviour of the difference between these two functions and find that $\sum_{n\leq x}\big( a(n)-P_1(n)\big) \sim \left(1-\fracπ{4}\right)\sum_{n\leq x} P_2(n)$, where $P_2(n)$ is the second largest divisor of $n$.
Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$, first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. This new setting gives us a parameter $q$ to vary and study how our bounds converge to 1 as it changes, and finite characteristic introduces some extra combinatorial structure that increases the tractibility of common questions in this area.
A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.