arXiv subjects

Andreas Weingartner

Publications and source records attributed to Andreas Weingartner.

23 records · Page 2Linked to original sources

arXiv2015

On the degrees of polynomial divisors over finite fields

We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than $m$. To that end, we first derive an improved estimate for the proportion of polynomials of degree $n$, all of whose non-constant divisors have degree greater than $m$. In the limit as $q \to \infty$, these results coincide with corresponding estimates related to the cycle structure of permutations.

math.NT↗

arXiv2015

Integers with large practical component

A positive integer $n$ is called practical if all integers between $1$ and $n$ can be written as a sum of distinct divisors of $n$. We give an asymptotic estimate for the number of integers $\le x$ which have a practical divisor $\ge y$.

math.NT↗

arXiv2015

Practical numbers and the distribution of divisors

An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.

math.NT↗

arXiv2010

The distribution functions of $σ(n)/n$ and $n/ϕ(n)$, II

Let $σ(n)$ be the sum of the positive divisors of $n$, and let $A(t)$ be the natural density of the set of positive integers $n$ satisfying $σ(n)/n \ge t$. We give an improved asymptotic result for $\log A(t)$ as $t$ grows unbounded. The same result holds if $σ(n)/n$ is replaced by $n/ϕ(n)$, where $ϕ(n)$ is Euler's totient function.

math.NT↗

arXiv2008

Zeros of Dirichlet series with periodic coefficients

Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(σ_1, σ_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $σ_1 < \Re s < σ_2$, $|\Im s | \le T$. We extend previous results of Laurinčikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_χ (s)$, where $P(s)$ is a Dirichlet polynomial and $L_χ(s)$ a Dirichlet L-function, then there exists an $η=η(a)>0$ such that for all $1/2 < σ_1 < σ_2 < 1+η$, we have $c_1 T \le N_a(σ_1, σ_2, T) \le c_2 T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, $σ_1$, and $σ_2$.

math.NT↗