Related papers: Integers with a divisor in (y,2y]

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: $\pi_{\nu}(x) \sim x \delta_{\nu}(x)$. Here we consider the restricted…

In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint…

A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…

The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove…

Each family $\mathcal{M}$ of means has a natural, partial order (point-wise order), that is $M \le N$ iff $M(x) \le N(x)$ for all admissible $x$. In this setting we can introduce the notion of interval-type set (a subset $\mathcal{I}…

Mader conjectured that for all k there is an integer d(k) such that every digraph of minimum outdegree at least d(k) contains a subdivision of a transitive tournament of order k. In this note we observe that if the minimum outdegree of a…

Let $N$ be any fixed positive integer and define \begin{align*} S_N(x)=\sum_{m, n \leq x}d(n^2+Nm^2), \end{align*} where $d(n)$ is the divisor function. We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov…

We establish a lower bound for the size of possible counterexamples of the Dixmier Conjecture. We prove that $B>15$, where $B$ is the minimum of the greatest common divisor of the total degrees of $P$ and $Q$, where $(P,Q)$ runs over the…

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization…

For a prime number $p>2$, we explain the construction of the difference divisors on the unitary Rapoport-Zink spaces of hyperspecial level and the GSpin Rapoport-Zink spaces of hyperspecial level associated to a minuscule cocharacter $\mu$…

The $i$-tuply $y$-densely divisible numbers were introduced by a Polymath project, as a weaker condition on the moduli than $y$-smoothness, in distribution estimates for primes in arithmetic progressions. We obtain the order of magnitude of…

One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all…

It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field…

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-\epsilon) for all N in a density one subset of the integers. If A is a…

We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-\gamma})^{-1} = 2.280...$ and $\gamma$ is…

We show that a modular unit on two copies of the upper half-plane is a Borcherds product if and only if its boundary divisor is a special boundary divisor. Therefore, we define a subspace of the space of invariant vectors for the Weil…

Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined over a number field $K$. In this paper, we study the denominators of the $x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of prime order,…

The index of codivisibility of a set of integers is the size of its largest subset with a common prime divisor. For large random samples of integers, the index of codivisibility is approximately normal.

We study a sorting machine consisting of two stacks in series where the first stack has the added restriction such that entries in the stack must be in decreasing order from top to bottom. We give the basis of the class of permutations that…