Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T20:09:00.816Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE NUMBER OF DIVISORS OF $n^{2}-1$

Part of: Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  02 October 2015

ADRIAN W. DUDEK
ADRIAN W. DUDEK*
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, Australia email adrian.dudek@anu.edu.au
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.

We prove an asymptotic formula for the sum $\sum _{n\leq N}d(n^{2}-1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum _{d\leq N}g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_{d}$ to the equation $x^{2}\equiv 1~(\text{mod}~d)$.

Keywords

divisor sumasymptotic estimatearithmetic functionsDiophantine quintuples

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 194 - 198
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Cipu, M., ‘Further remarks on Diophantine quintuples’, Acta Arith. 168 (2015), 201–219.Google Scholar
Cojocaru, A. C. and Murty, M. R., An Introduction to Sieve Methods and Their Applications (Cambridge University Press, Cambridge, 2006).Google Scholar
Dujella, A., ‘There are only finitely many Diophantine quintuples’, J. reine angew. Math. 566 (2004), 183–214.Google Scholar
Elsholtz, C., Filipin, A. and Fujita, Y., ‘On Diophantine quintuples and D (−1)-quadruples’, Monatsh. Math. 175(2) (2014), 227–239.CrossRefGoogle Scholar
Hooley, C., ‘On the number of divisors of quadratic polynomials’, Acta Math. 110(1) (1963), 97–114.Google Scholar
Ingham, A. E., ‘Some asymptotic formulae in the theory of numbers’, J. Lond. Math. Soc. (2) 1(3) (1927), 202–208.Google Scholar
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function (Oxford University Press, New York, 1986).Google Scholar
Trudgian, T. S., ‘Bounds on the number of Diophantine quintuples’, J. Number Theory 157 (2015), 233–249.Google Scholar