Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T00:43:07.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive arithmetic functions on intervals

Published online by Cambridge University Press:  24 October 2008

P. D. T. A. Elliott
P. D. T. A. Elliott
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

§1. A real-valued arithmetic function f is additive if it satisfies the relation f(ab) = f(a) + f(b) for all mutually prime positive integers a, b. In the present paper I establish three theorems concerning the value distribution of such functions on intervals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 163 - 179
Copyright
Copyright © Cambridge Philosophical Society 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Elliott, P. D. T. A.. Arithmetic Functions and Integer Products. Grundlehren der math. Wiss. 272 (Springer-Verlag, 1984).Google Scholar
[2]Elliott, P. D. T. A.. General asymptotic distributions for additive arithmetic functions. Math. Proc. Cambridge Philos. Soc. 79 (1976), 43–54.CrossRefGoogle Scholar
[3]Elliott, P. D. T. A.. Probabilistic Number Theory. (Two volumes.) Grundlehren der math. Wiss. 239, 240 (Springer-Verlag, 1979/1980).CrossRefGoogle Scholar
[4]Gnedenko, B. V. and Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables. Transl. from the Russian and annotated by K. L. Chung (Addison-Wesley, 1968).Google Scholar
[5]Halász, G.. On the distribution of additive arithmetic functions. Acta Arith. 27 (1975), 143–152.CrossRefGoogle Scholar
[6]Hildebrand, A.. Multiplicative functions in short intervals. Canadian J. Math. to appear.Google Scholar
[7]Hyers, D. H.. On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA, 27 (1941), 222–224.CrossRefGoogle ScholarPubMed
[8]Loéve, M.. Probability Theory (D. Van Nostrand, 1963).Google Scholar
[9]Lukacs, E.. Characteristic Functions (second edition) (Griffin, 1970).Google Scholar
[10]Ruzsa, I. Z.. The law of large numbers for additive functions. Studia Sci. Math. Hungar. 14 (1979), 247–253 (1982).Google Scholar
[11]Ruzsa, I. Z.. On the concentration of additive functions. Acta Math. Acad. Sci. Hungar. 36 (1980), 215–232.CrossRefGoogle Scholar
[12]Titchmarsh, E. C.. The Theory of the Riemann Zeta-Function (Clarendon Press, Oxford, 1951).Google Scholar