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Additive functions and special sets of integers

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

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.

Keywords

  • Additive Function
  • Asymptotic Formula
  • Prime Power
  • Normal Order
  • Saddle Point Method

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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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