On primitive abundant numbers

On the distribution of primitive abundant numbers

On Totient Abundant Numbers

In this note, we find an asymptotic formula for the counting function of the set of totient abundant numbers.

Νοτε Ον Consecutive Abundant Numbers

A positive integer N is called an abundant number if σ(N) > 2N, where σ (N) denotes the suns of the divisors of N including 1 and N. Abundant numbers have been recently investigated by Behrend, Chowla, Davenport , myself, and others ; it has been proved, for example, that they have a density greater than 0 . I prove now the following THEOREM. We can find two constants c l , c 2 such that , for ...

Variations on a Theorem of Davenport concerning Abundant Numbers

Let σ(n) = ∑ d|nd be the usual sum-of-divisors function. In 1933, Davenport showed that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u) := limx→∞(1/x) ∑ n≤x,n/σ(n)≤u1 exists for all u ∈ [0, 1] and varies continuously with u. We study the behaviour of the sums ∑n≤x,n/σ(n)≤u f (n) for certain complex-valued multiplicative functions f . Our results cover many of...

Abundant Numbers and the Riemann Hypothesis