arXiv subjects
Publications and source records attributed to Andreas Weingartner.
Buchstab's function $\omega(u)$ describes the distribution of integers without small prime factors. We establish numerically explicit upper and lower bounds for $\omega(u)$ that are easy to evaluate, without the need to solve the delay differential equation numerically.
Dickman's function $\rho(u)$ denotes the natural density of integers $n$ whose largest prime factor does not exceed $n^{1/u}$. We establish numerically explicit upper and lower bounds for $\rho(u)$, resulting in estimates with a relative error of less than $0.005/u^2$ for all $u\ge 5$. This allows for an approximate evaluation of $\rho(u)$ without the need to solve the delay differential equation numerically.
We establish a relationship between error terms appearing in estimates for the counting functions of smooth and rough numbers. We then apply this link to obtain an explicit upper bound for the error term in de Bruijn's approximation $\Lambda$ for the count of smooth numbers, from an explicit upper bound, due to Fan, for the error term in a variant of de Bruijn's estimate for the count of rough numbers.
A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions.
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 the count of these integers up to $x$, uniformly in $x$ and $y$, for every fixed natural number $i$.
We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-γ})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.
We derive asymptotic estimates for distribution functions related to the Schinzel-Szekeres function. These results are then used in three different applications: the longest simple path in the divisor graph, a problem of Erd\H{o}s about a sum of reciprocals, and the small sieve of Erd\H{o}s and Ruzsa.
We show that the sequence $(αn)_{n\in \mathcal{B}}$ is uniformly distributed modulo 1, for every irrational $α$, provided $\mathcal{B}$ belongs to a certain family of integer sequences, which includes the prime, almost prime, squarefree, practical, densely divisible and lexicographical numbers. We also give an estimate for the discrepancy if $α$ has finite irrationality measure.
We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of Margenstern.
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 $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
We use exponent pairs to establish the existence of many $x^a$-smooth numbers in short intervals $[x-x^b,x]$, when $a>1/2$. In particular, $b=1-a-a(1-a)^3$ is admissible. Assuming the exponent-pairs conjecture, one can take $b=(1-a)/2+ε$. As an application, we show that $[x-x^{0.4872},x]$ contains many practical numbers when $x$ is large.
We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers.
A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime $k$-tuples conjecture for practical numbers, proving the "correct" upper bound, and for pairs, improving on a lower bound of Melfi.
An integer $n\ge 1$ is said to be practical if every natural number $ m \le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $c x/\log x$, where $c$ is a constant. In this note we show that $c=1.33607...$.
We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.
We give an asymptotic estimate for the number of partitions of a set of $n$ elements, whose block sizes avoid a given set $\mathcal{S}$ of natural numbers. As an application, we derive an estimate for the number of partitions of a set with $n$ elements, which have the property that its blocks can be combined to form subsets of any size between $1$ and $n$.
Let $θ$ be an arithmetic function and let $\mathcal{B}$ be the set of positive integers $n=p_1^{α_1} \cdots p_k^{α_k}$, which satisfy $p_{j+1} \le θ( p_1^{α_1}\cdots p_{j}^{α_{j}})$ for $0\le j < k$. We show that $\mathcal{B}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of $\mathcal{B}$. When $θ(n)/n$ is non-decreasing, the set $\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1< d_2 < \ldots <d_{τ(n)}=n$ satisfy $d_{j+1} \le θ( d_j )$ for $1\le j <τ(n)$.
A positive integer $n$ is called $φ$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $φ$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.