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Additive functions and special sets of integers

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Book cover Number Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 938))

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Abstract

An additive function i is one that satisfies f(m.n)=f(m)+f(n) for positive integers m and n with (m,n)=1. A special set of integers S is any subset of the positive integers. We discuss the value distribution of f(n) for m ∈ S. Generalisations and extensions of the Hardy-Ramanujan results on normal order, the Turán-Kubilius inequality, and the Erdös-Kac theorem are obtained. Two kinds of special sets are considered. One class consists of sets S that are obtained as multiplicative semigroups generated by a prescribed collection of prime powers. The second class of sets are those in which the frequency of elements which are multiples of an integer d can be given in a convenient form in terms of d. This is a preliminary report of our recent researches in this area.

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Authors and Affiliations

  1. Department of Mathematics, The University of Michigan, 48109, ANN ARBOR, Michigan, USA

    Krishnaswami Alladi

  2. Matscience The Institute of Mathematical Sciences, 600 020, MADRAS, India

    Krishnaswami Alladi

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  1. Krishnaswami Alladi
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Krishnaswami Alladi

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© 1982 Springer-Verlag

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Alladi, K. (1982). Additive functions and special sets of integers. In: Alladi, K. (eds) Number Theory. Lecture Notes in Mathematics, vol 938. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097172

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  • DOI: https://doi.org/10.1007/BFb0097172

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11568-7

  • Online ISBN: 978-3-540-39279-8

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