Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T08:23:57.668Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  09 April 2009

Florian Luca
Florian Luca
Affiliation:
Mathematical Institute, UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico e-mail: fluca@matmor.unam.mx
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.

In 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.

Keywords

sum-of-divisors functionEuler functionnumber of prime divisors function

MSC classification

Secondary: 11A25: Arithmetic functions; related numbers; inversion formulas 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 149 - 164
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bateman, P. T., Erdős, P., Pomerance, C. and Straus, E. G., ‘The arithmetic mean of the divisors of an integer’, in: Analytic number theory (Philadelphia, PA, 1980), Lecture Notes in Math. 899 (Springer, Berlin, 1981) pp. 197–220.CrossRefGoogle Scholar
[2]Cooper, C. N. and Kennedy, R. N., ‘Chebyshev's inequality and natural density’, Amer. Math. Monthly 96 (1989), 118–124.CrossRefGoogle Scholar
[3]Elliott, P. D. T. A., Probabilistic number theory II (Springer, New York, 1980).CrossRefGoogle Scholar
[4]Erdős, P. and Pomerance, C., ‘On a theorem of Besicovitch: values of arithmetic functions that divide their arguments’, Indian J. Math. 32 (1990), 279–287.Google Scholar
[5]Luca, F., ‘On positive numbers n for which Ω(n) divides Fn’, Fibonacci Quart. 41 (2003), 365–371.Google Scholar
[6]Luca, F. and Pomerance, C., ‘On some problems of Mkowski-Schinzel and Erdős concerning the arithmetical functions φ and σ’, Colloq. Math. 92 (2002), 111–130.CrossRefGoogle Scholar
[7]Luca, F. and Shparlinski, I., ‘Arithmetical functions with linearly recurrent sequences’, preprint, 2002.Google Scholar
[8]Montgomery, H. L., Topics in multiplicative number theory, Lecture Notes in Math. 227 (Springer, New York, 1971).CrossRefGoogle Scholar
[9]Norton, K. K., ‘The number of restricted prime factors of an integer. I’, Illinois J. Math. 20 (1976), 681–705.CrossRefGoogle Scholar
[10]Norton, K. K., ‘The number of restricted prime factors of an integer. II’, Acta Math. 143 (1979), 9–38.CrossRefGoogle Scholar
[11]Rényi, A., ‘On the density of certain sequence of integers’, Acad. Serbe Sci. Publ. Inst. Math. 8 (1955), 157–162.Google Scholar
[12]Spiro, C., ‘How often is the number of divisors of n a divisor of n?’, J. Number Theory 21 (1985), 81–100.CrossRefGoogle Scholar
[13]Tenenbaum, G., Introduction to analytic and probabilistic number theory (Cambridge Univ. Press, Cambridge, 1995).Google Scholar
[14]Turán, P., ‘On a theorem of Hardy and Ramanujan’, J. London Math. Soc. 9 (1934), 274–276.CrossRefGoogle Scholar
[15]Wolke, D. and Zhan, T., ‘On the distribution of integers with a fixed number of prime factors’, Math. Z. 213 (1993), 133–144.CrossRefGoogle Scholar