Partition function (number theory): Difference between revisions

Revision as of 09:58, 18 January 2023

Revision as of 10:02, 18 January 2023

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:<math> \log p_A(n) \sim C \sqrt{\alpha n} </math>

and conversely if this asymptotic property holds for ''p''_''A''(''n'') then ''A'' has natural density α.{{sfn|Nathanson|2000|pp=475-85}} This result was stated, with a sketch of proof, by Erdős in 1942.<ref name=erdos42>{{cite journal | zbl=0061.07905 | last=Erdős | first=Pál | author-link=Paul Erdős | title=On an elementary proof of some asymptotic formulas in the theory of partitions | journal=Ann. Math. | series=(2) | volume=43 | pages=437–450 | year=1942 | issue=3 | doi=10.2307/1968802| jstor=1968802 }}</ref>{{sfn|Nathanson|2000|p=495}}

If ''A'' is a finite set, this analysis does not apply (the density of a finite set is zero). If ''A'' has ''k'' elements whose greatest common divisor is 1, then{{sfn|Nathanson|2000|pp=458-64}}