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Multiplicative functions at consecutive integers. II

Published online by Cambridge University Press:  24 October 2008

Adolf Hildebrand
Adolf Hildebrand
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, U.S.A.
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Extract

The global behaviour of multiplicative arithmetic functions has been extensively studied and is now well understood for a large class of multiplicative functions. In particular, Halász [5] completely determined the asymptotic behaviour of the means

for multiplicative functions g satisfying |g| ≤ 1, and gave necessary and sufficient conditions for the existence of the ‘mean value’

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 389 - 398
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Elliott, P. D. T. A.. Probabilistic Number Theory I (Springer-Verlag, 1979).Google Scholar
[2]Erdös, P.. On the distribution function of additive functions. Ann. of Math. (2) 47 (1946), 1–20.CrossRefGoogle Scholar
[3]Erdös, P., Pomerance, C. and Sarközy, A.. On locally repeated values of arithmetic functions. III. Proc. Amer. Math. Soc. 101 (1987), 1–7.CrossRefGoogle Scholar
[4]Friedlander, J. and Iwaniec, H.. On Bombieri's asymptotic sieve. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978), 719–756.Google Scholar
[5]Halász, G.. Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403.Google Scholar
[6]Halberstam, H. and Richert, H.-E.. Sieve Methods (Academic Press, 1974).Google Scholar
[7]Hildebrand, A.. Multiplicative functions at consecutive integers. Math. Proc. Cambridge Philos. Soc. 100 (1986), 229–236.Google Scholar
[8]Hildebrand, A.. An Erdös–Wintner theorem for differences of additive functions. Trans. Amer. Math. Soc, (to appear).Google Scholar
[9]Iwaniec, H.. Sieving limits. In Séminaire de Théorie des Nombres, Paris 1979–1980 (Birkhäuser, 1981), pp. 151–169.Google Scholar
[10]Kátai, I.. On a problem of P. Erdös. J. Number Theory 21 (1970), 1–6.Google Scholar
[11]Kátai, I.. Some problems in number theory. Studia Sci. Math. Hungar. 16 (1981), 289–295.Google Scholar
[12]Kátai, I.. Multiplicative functions with regularity properties. II. Acta Math. Acad. Sci. Hungar. 43 (1984), 105–130.Google Scholar
[13]Kuipers, L. and Niederreiter, H.. Uniform distribution of sequences (Wiley, 1974).Google Scholar
[14]Mauclaire, J.-L. and Murata, L.. On the regularity of arithmetic multiplicative functions. I. Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 438–440.Google Scholar
[15]Prachar, K.. Primzahlverteilung (Springer-Verlag, 1957).Google Scholar
[16]Wirsing, E.. A characterization of log n as an additive function. Sympos. Math. IV (1970), 45–57.Google Scholar
[17]Wirsing, E.. Private communication (October 1986).Google Scholar