Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T16:39:36.682Z Has data issue: false hasContentIssue false
  • English
  • Français

NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

Part of: Polynomials and matrices Elementary number theory

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Bellman, R., ‘Ramanujan sums and the average value of arithmetic functions’, Duke Math. J. 17 (1950), 159–168.Google Scholar
Cipu, M. and Trudgian, T., ‘Searching for Diophantine quintuples’, Acta Arith. 173(4) (2016), 365–382.Google Scholar
Deshouillers, J.-M. and Iwaniec, H., ‘An additive divisor problem’, J. Lond. Math. Soc. (2) 26(2) (1982), 1–14.Google Scholar
Dudek, A. W., ‘On the number of divisors of n 2 - 1’, Bull. Aust. Math. Soc. 93(2) (2016), 194–198.Google Scholar
Erdös, P., ‘On the sum ∑ k=1 x d (f (k))’, J. Lond. Math. Soc. (2) 27 (1952), 7–15.Google Scholar
Estermann, T., ‘Über die darstellung einer zahl als differenz von zwei produkten’, J. reine angew. Math. 164 (1931), 173–182.Google Scholar
Halberstam, H., ‘Four asymptotic formulae in the theory of numbers’, J. Lond. Math. Soc. 24 (1949), 13–21.Google Scholar
Hooley, C., ‘On the number of divisors of quadratic polynomials’, Acta Math. 110 (1963), 97–114.Google Scholar
Ingham, A. E., ‘Mean-value theorems in the theory of the Riemann zeta function’, Proc. Lond. Math. Soc. (2) 27 (1926), 273–300.Google Scholar
Ingham, A. E., ‘Some asymptotic formulae in the theory of numbers’, J. Lond. Math. Soc. 2 (1927), 202–208.Google Scholar
Lapkova, K., ‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729.Google Scholar
McKee, J., ‘On the average number of divisors of quadratic polynomials’, Math. Proc. Cambridge Philos. Soc. 117 (1995), 389–392.Google Scholar
McKee, J., ‘A note on the number of divisors of quadratic polynomials’, in: Sieve Methods, Exponential Sums, and their Applications in Number Theory, London Mathematical Society Lecture Note Series, 237 (Cambridge University Press, Cambridge, 1997), 275–281.Google Scholar
McKee, J., ‘The average number of divisors of an irreducible quadratic polynomial’, Math. Proc. Cambridge Philos. Soc. 126(1) (1999), 17–22.Google Scholar
Mordell, L. J., ‘On Mr. Ramanujan’s empirical expansions of modular functions’, Proc. Cambridge Philos. Soc. 19 (1917), 117–124.Google Scholar
Scourfield, E. J., ‘The divisors of a quadratic polynomial’, Proc. Glasg. Math. Soc. 5 (1961), 8–20.Google Scholar