Another teorem, in connection with (1), is the following: Denote for every positive real number \beta the number prodpa||np[\alpha\beta] by < n\beta>. Then for any \epsilon between 0 and 2, the set of integers n for which < d(n)2-\epsilon>/sigma(n) has asymptotic density 1, the set of n for which < d(n)2+\epsilon>/\sigma(n) has asymptotic density 0, and the set of n for which d(n)2/\sigma(n) has asymptotic desity ½. The proofs are long and complicated, with applications of results from various parts of number theory. To mention only a few: sieve methods, the generalized Erdös-Kac theorem and Tauberian theorems of Delange.
Classif.: * 11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes
Keywords: divisor function; sum of divisor function; arithmetic mean of divisors; asymptotic density
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag
|Books||Problems||Set Theory||Combinatorics||Extremal Probl/Ramsey Th.|
|Graph Theory||Add.Number Theory||Mult.Number Theory||Analysis||Geometry|
|Probabability||Personalia||About Paul Erdös||Publication Year||Home Page|