On a conjecture of Erdös about additive functions
Author:
Karl-Heinz Indlekofer
Journal:
Theor. Probability and Math. Statist. 89 (2014), 23-31
MSC (2010):
Primary 11N37, 11N60, 11K65
DOI:
https://doi.org/10.1090/S0094-9000-2015-00932-7
Published electronically:
January 26, 2015
MathSciNet review:
3235172
Full-text PDF Free Access
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Abstract:
For a real-valued additive function $f\colon \mathbb N \rightarrow \mathbb R$ and for each $n \in \mathbb N$ we define a distribution function \[ F_{n}(x):=\frac {1}{n}\#\{m\leq n\colon f(m)\leq x\}. \] In this paper we prove a conjecture of Erdös, which asserts that in order for the sequence $F_{n}$ to be (weakly) convergent, it is sufficient that there exist two numbers $a<b$ such that $\lim _{n\rightarrow \infty }(F_{n}(b)-F_{n}(a))$ exists and is positive.
The proof is based upon the use of the Stone–Čech compactification $\beta \mathbb N$ of $\mathbb N$ to mimic the behavior of an additive function as a sum of independent random variables.
References
- P. D. T. A. Elliott, A conjecture of Kátai, Acta Arith. 26 (1974/75), 11–20. MR 354599, DOI https://doi.org/10.4064/aa-26-1-11-20
- P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
- P. Erdös, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1–20. MR 15424, DOI https://doi.org/10.2307/1969031
- Paul Erdös, Some remarks and corrections to one of my papers, Bull. Amer. Math. Soc. 53 (1947), 761–763. MR 21030, DOI https://doi.org/10.1090/S0002-9904-1947-08868-6
- Adolf Hildebrand, Additive and multiplicative functions on shifted primes, Proc. London Math. Soc. (3) 59 (1989), no. 2, 209–232. MR 1004429, DOI https://doi.org/10.1112/plms/s3-59.2.209
- Adolf Hildebrand, A note on the convergence of sums of independent random variables, Ann. Probab. 20 (1992), no. 3, 1204–1212. MR 1175258
- K.-H. Indlekofer, A new method in probabilistic number theory, Probability theory and applications, Math. Appl., vol. 80, Kluwer Acad. Publ., Dordrecht, 1992, pp. 299–308. MR 1211915
- Karl-Heinz Indlekofer, New approach to probabilistic number theory—compactifications and integration, Probability and number theory—Kanazawa 2005, Adv. Stud. Pure Math., vol. 49, Math. Soc. Japan, Tokyo, 2007, pp. 133–170. MR 2405602, DOI https://doi.org/10.2969/aspm/04910133
- I. Kátai, On distribution of arithmetical functions on the set prime plus one, Compositio Math. 19 (1968), 278–289. MR 244181
- Fanchao Kong and Qihe Tang, A theorem on the convergence of sums of independent random variables, Acta Math. Sci. Ser. B (Engl. Ed.) 21 (2001), no. 3, 331–338. MR 1849532, DOI https://doi.org/10.1016/S0252-9602%2817%2930419-8
- Fanchao Kong and Qihe Tang, Notes on Erdős’ conjecture, Acta Math. Sci. Ser. B (Engl. Ed.) 20 (2000), no. 4, 533–541. MR 1805022, DOI https://doi.org/10.1016/S0252-9602%2817%2930664-1
- P. Lévy, Sur les séries dont les termes sont des variables éventuellement indépendents, Studia Math. 3 (1931), 119–155.
- Michel Loève, Probability theory. I, 4th ed., Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 45. MR 0651017
- Russell C. Walker, The Stone-Čech compactification, Springer-Verlag, New York-Berlin, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. MR 0380698
References
- P. D. T. A. Elliott, A conjecture of Kátai, Acta Arith. XXVI (1974), 11–20. MR 0354599 (50:7077)
- P. D. T. A. Elliott, Probabilistic Number Theory I/II, Springer, New York, 1978/1980. MR 551361 (82h:10002a)
- P. Erdös, On the distribution function of additive functions, Ann. Math. 47 (1946), 1–20. MR 0015424 (7:416c)
- P. Erdös, Some remarks and corrections to one of my papers, Bull. Amer. Math. Soc. 53 (1947), 761–763. MR 0021030 (9:12e)
- A. Hildebrand, Additive and multiplicative functions on shifted primes, Proc. London Math. Soc. 59(2) (1989), 209–232. MR 1004429 (90i:11107)
- A. Hildebrand, A note on the convergence of sums of independent random variables, Ann. Probab. 20 (1992), 1204–1212. MR 1175258 (93g:60047)
- K.-H. Indlekofer, A new method in probabilistic number theory, Probab. Theory Appl., Math. Appl. 80 (1992), 299–308. MR 1211915 (94e:11091)
- K.-H. Indlekofer, New approach to probabilistic number theory — compactifications and integration, Adv. Studies Pure Math. 49 (2005), 133–170. MR 2405602 (2009d:11119)
- I. Kátai, On the distribution of arithmetical functions on the set prime plus one, Composito Math. 19 (1968), 278–289. MR 0244181 (39:5498)
- F. Kong and Q. Tang, A theorem on the convergence of sums of independent random variables, Math. Acta Sci. 21 B (2001), 331–338. MR 1849532 (2002f:60052)
- F. Kong and Q. Tang, Notes on Erdös’ conjecture, Math. Acta Sci. 20(4) (2000), 533–541. MR 1805022 (2001k:60032)
- P. Lévy, Sur les séries dont les termes sont des variables éventuellement indépendents, Studia Math. 3 (1931), 119–155.
- M. Loève, Probability theory I, Springer, Heidelberg–New York, 1977. MR 0651017 (58:31324a)
- R. Walker, The Stone–Čech Compactification, Springer, Heidelberg–New York, 1974. MR 0380698 (52:1595)
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Additional Information
Karl-Heinz Indlekofer
Affiliation:
Department of Mathematics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany
Email:
k-heinz@math.upb.de
Keywords:
Probabilistic number theory,
additive functions
Received by editor(s):
January 29, 2013
Published electronically:
January 26, 2015
Article copyright:
© Copyright 2015
American Mathematical Society