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NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

Part of: Elementary number theory Polynomials and matrices

Published online by Cambridge University Press:  15 August 2018

ADRIAN W. DUDEK ,
ŁUKASZ PAŃKOWSKI Open the ORCID record for ŁUKASZ PAŃKOWSKI [Opens in a new window]  and
VICTOR SCHARASCHKIN
ADRIAN W. DUDEK
Affiliation:
Cronulla NSW 2230, Australia email awdudek@gmail.com
ŁUKASZ PAŃKOWSKI*
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland email lpan@amu.edu.pl
VICTOR SCHARASCHKIN
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, QLD 4072, Australia email vscharaschkin@gmail.com
*
✉lpan@amu.edu.pl
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Abstract

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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 $\sum _{n\leq x}d(n)d(n+v)$.

Keywords

divisor functionreducible quadratic polynomialsadditive divisor problem

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11C08: Polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 1 - 9
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author was partially supported by the Grant no. 2016/23/D/ST1/01149 from the National Science Centre.

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