Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T04:22:02.719Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Daboussi, H. and Delange, H., On a theorem of P. D. T. A. Elliott on multiplicative functions, J. London Math. Soc. 14 (1976), 345–356.Google Scholar
2. Elliott, P. D. T. A., Probabilistic number theory, Grundlehren vols. 239, 240 (Springer Verlag, Berlin, New York, 1979, 1980).Google Scholar
3. Elliott, P. D. T. A., On connections between the Turán-Kubilius inequality and the Large Sieve: Some applications, Proceedings of Symposia in Pure Math. 24 (Amer. Math. Soc., Providence, 1973), 77–82.Google Scholar
4. Elliott, P. D. T. A., A mean-value theorem for multiplicative functions, Proc. London Math. Soc. 31 (1975), 418–438.Google Scholar
5. Elliott, P. D. T. A. and Erdös, P., Additive arithmetic functions bounded by monotone functions on thin sets, to appear in the Annals of Budapest University.Google Scholar
6. Erdös, P. and Kac, M., On the Gaussian law of errors in the theory of additive functions, Proc. Nat. Acad. Sci., U.S.A. 25 (1939), 206–207.Google Scholar
7. Forti, M. and Viola, C., On Large Sieve type estimates for the Dirichlet series operator, Proceedings of Symposia in Pure Math. 24 (Amer. Math. Soc, Providence, 1973), 31–49.Google Scholar
8. Hall, R. R., Halving an estimate obtained from Selbergs upper bound method, Acta Arithmetica 25 (1974), 347–351.Google Scholar
9. Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
10. Kubilius, J., Probabilistic methods in the theory of numbers, Amer. Math. Soc. Translations of Math. Monographs 11 (1964).Google Scholar
11. Turán, P., On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274–276.Google Scholar
12. Turán, P., Über einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan, J. London Math. Soc. 11 (1936), 125–133.Google Scholar