Normal order of an arithmetic function: Difference between revisions

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In [[mathematics]], in the field of [[number theory]], the '''normal order of an arithmetic function''' is some simpler or better-understood function which "usually" takes the same or closely approximate values.
In [[mathematics]], in the field of [[number theory]], the '''normal order of an arithmetic function''' is some simpler or better-understood function which "usually" takes the same or closely approximate values.


Let ''f'' be a function on the [[natural number]]s.  We say that the ''normal order'' of ''f'' is ''g'' if for every &epsilon > 0, the inequalities
Let ''f'' be a function on the [[natural number]]s.  We say that the ''normal order'' of ''f'' is ''g'' if for every ε > 0, the inequalities


:<math> (1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n) \, </math>
:<math> (1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n) \, </math>
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==Examples==
==Examples==
* The [[Hardy–Ramanujan theorem]]: the normal order of &omega;(''n''), the number of distinct [[prime factor]]s of ''n'', is log(log(''n''));
* The [[Hardy–Ramanujan theorem]]: the normal order of &omega;(''n''), the number of distinct [[prime factor]]s of ''n'', is log(log(''n''));
* The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2) log log(''n'').  
* The normal order of log(''d''(''n'')), where ''d''(''n'') is the [[number of divisors function|number of divisors]] of ''n'', is log(2) log log(''n'').


==See also==
==See also==
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* {{cite journal|author=G.H. Hardy| authorlink=G. H. Hardy| coauthors=S. Ramanujan|title=The normal number of prime factors of a number |journal= Quart. J. Math. |volume= 48  |year=1917|pages= 76–92}}
* {{cite journal|author=G.H. Hardy| authorlink=G. H. Hardy| coauthors=S. Ramanujan|title=The normal number of prime factors of a number |journal= Quart. J. Math. |volume= 48  |year=1917|pages= 76–92}}
* {{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=E.M. Wright | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | pages=473 | year=2008 | isbn=0-19-921986-5 }}  
* {{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=E.M. Wright | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | pages=473 | year=2008 | isbn=0-19-921986-5 }}  
* {{cite book | title=Introduction to Analytic and Probabilistic Number Theory | author=Gérald Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | pages=299-324 | year=1995 | isbn=0-521-41261-7 }}
* {{cite book | title=Introduction to Analytic and Probabilistic Number Theory | author=Gérald Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | pages=299-324 | year=1995 | isbn=0-521-41261-7 }}[[Category:Suggestion Bot Tag]]

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In mathematics, in the field of number theory, the normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values.

Let f be a function on the natural numbers. We say that the normal order of f is g if for every ε > 0, the inequalities

hold for almost all n: that is, if the proportion of nx for which this does not hold tends to 0 as x tends to infinity.

It is conventional to assume that the approximating function g is continuous and monotone.

Examples

See also

References

  • G.H. Hardy; S. Ramanujan (1917). "The normal number of prime factors of a number". Quart. J. Math. 48: 76–92.