scholarly journals On the number of divisors of quadratic polynomials

1963 ◽  
Vol 110 (0) ◽  
pp. 97-114 ◽  
Author(s):  
Christopher Hooley
Keyword(s):  
Quadratic Polynomials ◽  
Number Of Divisors

scholarly journals NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

2018 ◽  
Vol 99 (1) ◽  
pp. 1-9
Author(s):  
ADRIAN W. DUDEK ◽  
ŁUKASZ PAŃKOWSKI ◽  
VICTOR SCHARASCHKIN
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Tauberian Theorem ◽  
Fixed Integer ◽  
Quadratic Polynomials ◽  
Number Of Divisors ◽  
The Relationship

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

On the average number of divisors of quadratic polynomials

1995 ◽  
Vol 117 (3) ◽  
pp. 389-392 ◽  
Author(s):  
James Mckee
Keyword(s):  
Quadratic Polynomials ◽  
Integer Coefficients ◽  
Number Of Divisors ◽  
Image Position

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

scholarly journals Distribution of polynomials with cycles of a given multiplier

2011 ◽  
Vol 201 ◽  
pp. 23-43 ◽  
Author(s):  
Giovanni Bassanelli ◽  
François Berteloot
Keyword(s):  
Quadratic Polynomials ◽  
Bifurcation Locus

AbstractIn the space of degreedpolynomials, the hypersurfaces defined by the existence of a cycle of periodnand multipliereiθare known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

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