THE ERROR TERM IN THE COUNT OF ABUNDANT NUMBERS
Mitsuo Kobayashi, Paul Pollack · Mathematika · 2014
A natural number is called abundant if the sum of the proper divisors of exceeds . For example, 12 is abundant, since . In 1929, Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. In other words, if denotes the count of abundant numbers belonging to the interval , does tend to a limit? Four years later, Davenport answered Bessel-Hagen's question in the affirmative. Calling this density , it is now known that , so that just under one in four numbers are abundant. We show that for all large . We also study the behavior of the corresponding error term for the count of so-called -abundant numbers.