Representation functions avoiding integers with density zero

2022 ◽  
Vol 102 ◽  
pp. 103490
Author(s):  
Jin-Hui Fang
Keyword(s):  
Density Zero

s -Maximal Nonbases of Density Zero

1977 ◽  
Vol s2-15 (1) ◽  
pp. 29-34 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Density Zero

scholarly journals On a Theorem of Niven

1979 ◽  
Vol 22 (1) ◽  
pp. 113-115
Author(s):  
R. Sita Rama Chandra Rao ◽  
G. Sri Rama Chandra Murty
Keyword(s):  
Density Zero

In [4], Niven proved that the set A of integers for all s ≥ l and all n ≥ 1 has density zero, being the sum of the sth powers of all positive divisors of n. However his argument contains a mistake (see Remark 1). In this paper we give a proof of Niven's result and establish several related results, one of which generalizes a result of Dressier (See Theorem 3 and Remark 2).

Nonbases of Density Zero not Contained in Maximal Nonbases

1977 ◽  
Vol s2-15 (3) ◽  
pp. 403-405 ◽  
Author(s):  
Paul Erdős ◽  
Melvyn B. Nathanson
Keyword(s):  
Density Zero

scholarly journals Ordinary reduction of K3 surfaces

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Zarhin
Keyword(s):  
Number Field ◽  
Field Extension ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Zero Set ◽  
Algebraic Field ◽  
Density Zero ◽  
Archimedean Place ◽  
Ordinary Reduction

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

On the local behavior of the order of appearance in the Fibonacci sequence

2014 ◽  
Vol 10 (04) ◽  
pp. 915-933 ◽  
Author(s):  
Florian Luca ◽  
Carl Pomerance
Keyword(s):  
Positive Integer ◽  
Rational Function ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Asymptotic Density ◽  
Infinitely Many Solutions ◽  
Local Behavior ◽  
Sieve Methods ◽  
Density Zero

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the k th Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

scholarly journals On f(n) modulo Ω(n) and ω(n) when f is a polynomial

2004 ◽  
Vol 77 (2) ◽  
pp. 149-164 ◽  
Author(s):  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Density Zero ◽  
Prime Factors ◽  
Nonzero Polynomial

AbstractIn this paper we show that if f (X) ∈; Z [X ] is a nonzero polynomial, then ω(n)/f(n) holds only on a set of n of asymptotic density zero, where for a positive integer n the number ω(n) counts the number of distinct prime factors ofn.

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