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On a problem of hardy and ramanujan

Published online by Cambridge University Press:  26 February 2010

P. D. T. A. Elliott
P. D. T. A. Elliott
Affiliation:
Boulder, Colorado, U.S.A.
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Extract

Some sixty years ago Hardy and Ramanujan [6]introduced the notion of normal order of an arithmetic function.

A real-valued arithmetic function f)n) is said to have a normal order if there is a function g(n), which is non-negative and non-decreasing for all sufficiently large integers n, so that, for each fixed ε > 0, the integers n, for which the inequality

is satisfied, have asymptotic density zero. Thus, in this certain sense, f(n) behaves almost surely like g(n). We say that f(n) has the normal order g(n). In their original paper Hardy and Ramanujan asked that the function g(n) be “elementary”, but this is a requirement that subsequent researchers have dropped.

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Type
Research Article
Information
Mathematika , Volume 23 , Issue 1 , June 1976, pp. 10 - 17
Copyright
Copyright © University College London 1976

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