Abstract
Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set 
of generalized integers associated with a set 
of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,
Assume that the distribution function N(x) of satisfies
with γ>γ0, where ρ1<ρ2<···<ρ
m
are constants with ρ
m
≥1 and A1,···,A
m
are real constants with A
m
>0. Also, assume that the Chebyshev function ψ(x) of satisfies
with M>M0. Then the asymptotic
implies
where τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bateman, P.T., Diamond, H.G.: Analytic Number Theory: An Introductory Course. World Scientific Pub. Co., Singapore, 2004
Bateman, P.T., Diamond, H.G.: Asymptotic distribution of Beurling’s generalized prime numbers. Studies in Number Theory, Vol. 6, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, N.J., 1969, pp. 152–210
Beurling, A.: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I. Acta Math. 68, 255–291 (1937)
Diamond, H., Montgomery, H., Vorhauer, U.: Beurling primes with large oscillation. 2003. To appear in Math. Ann., Digital Object Identifier (DOI) 10.1007/s00208-005-0638-2 (2005)
Elliott, P.D.T.A.: Probabilistic Number Theory, I: Mean Value Theorems. Grund. der Math. Wiss. 239, Springer, New York, Heidelberg, Berlin, 1979
Halász, G.: Über die Mittewerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hung. 19, 365–403 (1968)
Indlekofer, K.-H., Wagner, R., Kátai, I.: A comparative result for multipicative functions. Liet. Matem. rink 41, 183–201 (2001)
Levin, B., Timofeev, E. Teorema sravnenija dlja multiplikativnyh funkcij. Acta Arith. XLII, 21–47 (1982)
Nyman, B.: A general prime number theorem. Acta Math. 81, 299–307 (1949)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. 4th edition, Cambridge Univ. Press, 1952
Zhang, W.-B.: A generalization of Halász’s theorem to Beurling’s generalized integers and its application. Illinois J. Math. 31, 645–664 (1987)
Zhang, W.-B.: Mean-value theorems for multiplicative functions on Beurling’s generalized integers. (unpublished)
Zhang, W.-B.: Chebyshev type estimates for Beurling generalized prime numbers. II. Trans. Am. Math. Soc. 337, 651–675 (1993)
Zhang, W.-B.: Mean-value theorems for multiplicative functions on additive arithmetic semigroups via Halász’s method. Monatshefte für Math. 138, 319–353 (2003)
Zhang, W.-B.: A general theorem for Beurling’s generalized primes. (to appear)
Author information
Authors and Affiliations
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/s00209-008-0426-2.
Rights and permissions
About this article
Cite this article
Zhang, WB. Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I). Math. Z. 251, 359–391 (2005). https://doi.org/10.1007/s00209-005-0807-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-005-0807-8