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Transactions of the American Mathematical Society

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An Erdős-Wintner theorem for differences of additive functions


Author: Adolf Hildebrand
Journal: Trans. Amer. Math. Soc. 310 (1988), 257-276
MSC: Primary 11K65; Secondary 11N60
MathSciNet review: 965752
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Abstract: An Erdös-Wintner type criterion is given for the convergence of the distributions $ {D_x}(z) = {[x]^{ - 1}}\char93 \{ 1 \leqslant n \leqslant x:\,f(n + 1) - f(n) \leqslant z\} $, where $ f$ is a real-valued additive function. A corollary of this result is that an additive function $ f$, for which $ f(n + 1) - f(n)$ tends to zero on a set of density one, must be of the form $ f = \lambda \log$ for some constant $ \lambda $. This had been conjectured by Erdős.


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  • [1] Gutti Jogesh Babu, Probabilistic methods in the theory of arithmetic functions, Macmillan Lectures in Mathematics, vol. 2, Macmillan Co. of India, Ltd., New Delhi, 1978. MR 515880
  • [2] Hubert Delange, Sur les fonctions arithmétiques multiplicatives, Ann. Sci. École Norm. Sup. (3) 78 (1961), 273–304 (French). MR 0169829
  • [3] P. D. T. A. Elliott, On the differences of additive arithmetic functions, Mathematika 24 (1977), no. 2, 153–165. MR 0472734
  • [4] P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361
  • [5] Peter D. T. A. Elliott, Probabilistic number theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 240, Springer-Verlag, Berlin-New York, 1980. Central limit theorems. MR 560507
  • [6] P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 272, Springer-Verlag, New York, 1985. MR 766558
  • [7] P. Erdös, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1–20. MR 0015424
  • [8] P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738–742. MR 0002374
  • [9] Paul Erdös and Aurel Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713–721. MR 0000247
  • [10] Adolf Hildebrand, Multiplicative functions at consecutive integers, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 2, 229–236. MR 848849, 10.1017/S0305004100066056
  • [11] Adolf Hildebrand, Multiplicative functions in short intervals, Canad. J. Math. 39 (1987), no. 3, 646–672. MR 905750, 10.4153/CJM-1987-032-6
  • [12] A. A. \cyr{K}aratsuba, Osnovy analiticheskoi teorii chisel, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0439767
  • [13] I. Kátai, On a problem of P. Erdős, J. Number Theory 2 (1970), 1–6. MR 0250991
  • [14] J. Kubilius, Probabilistic methods in the theory of numbers, Translations of Mathematical Monographs, Vol. 11, American Mathematical Society, Providence, R.I., 1964. MR 0160745
  • [15] Eduard Wirsing, Characterisation of the logarithm as an additive function, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N. Y., 1969) Amer. Math. Soc., Providence, R. I., 1971, pp. 375–381. MR 0412129

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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0965752-X
Article copyright: © Copyright 1988 American Mathematical Society