Abstract
To better understand the distribution of gaps between k-free numbers, Erdős posed the problem of establishing an asymptotic formula for the sum of the powers of the lengths of the gaps between k-free numbers. This paper generalizes the problem of Erdős by considering moments of gaps between positive integers m for which f(m) is k-free. Here, f(x) denotes an irreducible polynomial with integer coeficients with some necessary conditions imposed on it. Some results in this general setting are obtained that are analogous to those that have been obtained for the original problem of Erdős.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. ErdŐs, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103–109.
P. ErdŐs, Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), 416–425.
M. Filaseta, On the distribution of gaps between squarefree numbers, Mathematika 40 (1993), 88–101.
M. Filaseta, Short interval results for k-free values of irreducible polynomials, Acta Arith. 64 (1993), 249–270.
S. W. Graham, Moments of gaps between k-free numbers, Journal of Number Theory 44 (1993), 105–117.
C. Hooley, On the power-free values of polynomials, Mathematika 14 (1967), 21–26.
C. Hooley, On the distribution of square-free numbers, Canad. J. Math. 25 (1973), 1216–1223.
M. N. Huxley, Moments of differences between square-free numbers, in: Sieve Methods, Exponential Sums, and their Applications in Number Theory (eds. G. R. H. Greaves, G. Harman, and M. N. Huxley), Cambridge Univ. Press, Cambridge, 1996/7, 187–204.
M. N. Huxley and M. Nair, Powerfree values of polynomials III, Proc. London Math. Soc. 41 (1980), 66–82.
L. Mirsky, Arithmetical pattern problems related to divisibility by rth powers, Proc. London Math. Soc. 50 (1949), 497–508.
T. Nagell, Zur Arithmetik der Polynome, Abh. Math. Sem. Hamburg. Univ. 1 (1922), 179–184.
M. Nair, Power-free values of polynomials, Mathematika 23 (1976), 159–183.
M. Nair, Power-free values of polynomials II, Proc. London Math. Soc. (3) 38 (1979), 353–368.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beasley, B., Filaseta, M. A distribution problem for powerfree values of irreducible polynomials. Periodica Mathematica Hungarica 42, 123–144 (2001). https://doi.org/10.1023/A:1015204825565
Issue Date:
DOI: https://doi.org/10.1023/A:1015204825565