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A class of additive arithmetic functions with asymptotically normal distribution

  • P. D. T. A. Elliott (a1)

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Let f(n) be a real-valued additive arithmetic function. For each positive rational integer n set

For the duration of this paper we shall assume that as n → ∞ both

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1. Barban, M. B., “On a theorem of J. P. Kubilius”, Izv. Akad. Nauk. USSR. Ser. Fiz. Mat. Nauk., 1961 (5), 39.
2. Erdős, P. and Kac, M., “The Gaussian law of errors in the theory of additive number-theoretic functions”, Amer. J. Math., 62 (1940), 738742.
3. Esseen, C. G., “Fourier Analysis of distribution functions. A mathematical study of the Laplace- Gaussian law”, Ada Math., 77 (1945), 1125.
4. Hálasz, G., “On the distribution of additive and the mean values of multiplicative arithmetic functions”, Studia Scient. Math. Hungar., 6 (1971), 211233.
5. Kubilius, J., “A class of additive number-theoretic functions with asymptotically normal distribution”, Naucn. Trudy Fiz-Tehn. Inst. Akad. Nauk. Lit. SSSR, 2 (1956), 515.
6. Kubilius, J., Probabilistic methods in the theory of numbers (American Math. Soc. Translations, No. 11).
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