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High-Power Analogues of the Turán-Kubilius Inequality, and an Application to Number Theory

Published online by Cambridge University Press:  20 November 2018

P. D. T. A. Elliott
P. D. T. A. Elliott*
Affiliation:
University of Colorado, Boulder, Colorado
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An arithmetic function ƒ(n) is said to be additive if it satisfies ƒ(ab) = ƒ(a) + ƒ(b) whenever a and b are coprime integers. For such a function we define

A standard form of the Turán-Kubilius inequality states that

(1)

holds for some absolute constant c1, uniformly for all complex-valued additive arithmetic functions ƒ (n), and real x ≧ 2. An inequality of this type was first established by Turán [11], [12] subject to some side conditions upon the size of │ƒ(pm)│. For the general inequality we refer to [10].

This inequality, and more recently its dual, have been applied many times to the study of arithmetic functions. For an overview of some applications we refer to [2]; a complete catalogue of the applications of the inequality (1) would already be very large. For some applications of the dual of (1) see [3], [4], and [1].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 893 - 907
Copyright
Copyright © Canadian Mathematical Society 1980

References

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