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:<math> \log p_A(n) \sim C \sqrt{\alpha n} </math>
with <math>C = \pi\sqrt\frac23</math>
and conversely if this asymptotic property holds for ''p''<sub>''A''</sub>(''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></ref>{{sfn|Nathanson|2000|p=495}}
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