Jordan's totient function: Difference between revisions
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In [[number theory]], '''Jordan's totient function''' <math>J_k(n)</math> of a [[positive integer]] ''n'', named after [[Camille Jordan]], is defined to be the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a [[coprime]] (''k'' + 1)-tuple together with ''n''. This is a generalisation of Euler's [[totient function]], which is ''J''<sub>1</sub>. | In [[number theory]], '''Jordan's totient function''' <math>J_k(n)</math> of a [[positive integer]] ''n'', named after [[Camille Jordan]], is defined to be the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a [[coprime]] (''k'' + 1)-tuple together with ''n''. This is a generalisation of Euler's [[totient function]], which is ''J''<sub>1</sub>. | ||
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* {{cite book | author=L. E. Dickson | authorlink=Leonard Eugene Dickson | title=History of the Theory of Numbers I | year=1919, repr.1971 | publisher=Chelsea | isbn=0-8284-0086-5 | page=147 }} | * {{cite book | author=L. E. Dickson | authorlink=Leonard Eugene Dickson | title=History of the Theory of Numbers I | year=1919, repr.1971 | publisher=Chelsea | isbn=0-8284-0086-5 | page=147 }} | ||
*{{cite book | title=Problems in Analytic Number Theory | author=M. Ram Murty | authorlink=M. Ram Murty | volume=206 | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | year=2001 | isbn=0387951431 | page=11 }} | *{{cite book | title=Problems in Analytic Number Theory | author=M. Ram Murty | authorlink=M. Ram Murty | volume=206 | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | year=2001 | isbn=0387951431 | page=11 }} | ||
Latest revision as of 16:05, 29 October 2008
In number theory, Jordan's totient function of a positive integer n, named after Camille Jordan, is defined to be the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. This is a generalisation of Euler's totient function, which is J1.
Definition
Jordan's totient function is multiplicative and may be evaluated as
Properties
- .
- The average order of Jk(n) is c nk for some c.
References
- L. E. Dickson (1919, repr.1971). History of the Theory of Numbers I. Chelsea. ISBN 0-8284-0086-5.
- M. Ram Murty (2001). Problems in Analytic Number Theory. Springer-Verlag. ISBN 0387951431.