Results 11 to 20 of about 312 (25)

AN AVERAGE THEOREM FOR TUPLES OF k‐FREE NUMBERS IN ARITHMETIC PROGRESSIONS

Mathematika, Volume 67, Issue 1, Page 1-35, January 2021., 2021
Abstract In the spirit of the Hooley–Montgomery refinement of the Barban–Davenport‐Halberstam theorem, we obtain an asymptotic formula for the variance associated with tuples of k‐free numbers in arithmetic progressions.
Tomos Parry
wiley   +1 more source

On a thin set of integers involving the largest prime factor function

International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 19, Page 1185-1192, 2003., 2003
For each integer n ≥ 2, let P(n) denote its largest prime factor. Let S : = {n ≥ 2 : n does not divide P(n)!} and S(x) : = #{n ≤ x : n ∈ S}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x) = O(x/logx). Recently, Akbik (1999) proved that S(x)=O(x exp{−(14/)logx}).
Jean-Marie De Koninck, Nicolas Doyon
wiley   +1 more source

On a density problem of Erdös

International Journal of Mathematics and Mathematical Sciences, Volume 22, Issue 3, Page 655-658, 1999., 1999
For a positive integer n, let P(n) denotes the largest prime divisor of n and define the set: 𝒮(x) = 𝒮 = {n ≤ x : n does not divide P(n)!}. Paul Erdös has proposed that |S| = o(x) as x → ∞, where |S| is the number of n ∈ S. This was proved by Ilias Kastanas. In this paper we will show the stronger result that .
Safwan Akbik
wiley   +1 more source

On curves over finite fields with Jacobians of small exponent

, 2008
We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1.
Ford, Kevin, Shparlinski, Igor
core   +1 more source

On the distribution of powered numbers

Open Mathematics
Asymptotic formulae are established for the number of natural numbers mm with largest square-free divisor not exceeding mϑ{m}^{{\vartheta }}, for any fixed positive parameter ϑ{\vartheta }. Related counting functions are also considered.
Brüdern Jörg, Robert Olivier
doaj   +1 more source

Multiplicative inverses in short intervals

, 2012
We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.Comment: 7 ...
Baier, Stephan
core   +1 more source

Practical numbers and the distribution of divisors

, 2015
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.
Weingartner, Andreas
core   +1 more source

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

, 2018
A composite positive integer $n$ has the Lehmer property if $\phi(n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this note we shall prove that if $n$ has the Lehmer property, then $n\leq 2^{2^{K}}-2^{2^{K-1}}$, where $K$ is the number ...
Burek, Dominik, Żmija, Błażej
core   +1 more source

Detecting mixedness of qutrit systems using the uncertainty relation

, 2013
We show that the uncertainty relation as expressed in the Robertson-Schrodinger generalized form can be used to detect the mixedness of three-level quantum systems in terms of measureable expectation values of suitably chosen observables when prior ...
A. S. Majumdar, E. Schrodinger, S. Mal, T. Pramanik   +3 more
core   +1 more source

The density of twins of $k$-free numbers

, 2014
For $k \geq 2$, we consider the number $A_k(Z)$ of positive integers $n \leq Z$ such that both $n$ and $n+1$ are $k$-free. We prove an asymptotic formula $A_k(Z) = c_k Z + O(Z^{14/(9k)+\epsilon})$, where the error term improves upon previously known ...
Dietmann, Rainer, Marmon, Oscar
core   +1 more source