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THE ERROR TERM IN THE COUNT OF ABUNDANT NUMBERS
Author(s) -
Kobayashi Mitsuo,
Pollack Paul
Publication year - 2014
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579313000235
Subject(s) - mathematics , term (time) , natural number , bessel function , interval (graph theory) , limit (mathematics) , set (abstract data type) , combinatorics , natural density , statistics , mathematical analysis , computer science , programming language , physics , quantum mechanics
A natural number n is called abundant if the sum of the proper divisors of n exceeds n . For example, 12 is abundant, since 1 + 2 + 3 + 4 + 6 = 16 . In 1929, Bessel‐Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if A ( x ) denotes the count of abundant numbers belonging to the interval [ 1 , x ] , does A ( x ) / x tend to a limit? Four years later, Davenport answered Bessel‐Hagen's question in the affirmative. Calling this density Δ , it is now known that 0.24761 < Δ < 0.24766 , so that just under one in four numbers are abundant. We show that A ( x ) − Δ x < x / exp ( ( log x ) 1 / 3 ) for all large x . We also study the behavior of the corresponding error term for the count of so‐called α ‐ abundant numbers .