# zbMATH — the first resource for mathematics

Bounds for the density of abundant integers. (English) Zbl 0923.11127
An integer $$n$$ is called abundant if the sum of the divisors of $$n$$ is at least $$2n$$. It is known that abundant numbers have natural density $$d$$ and that $$0.244<d<0.291$$. The author answers a question by H. Cohen, showing that $$d<1/4$$. Precisely, using a method of Behrend the authors shows that $$0.2474<d<0.2480$$.

##### MSC:
 11N25 Distribution of integers with specified multiplicative constraints 11A25 Arithmetic functions; related numbers; inversion formulas
Full Text:
##### References:
 [1] Behrend F., Sitzungsber. Preuss. Akad. Wiss. pp 280– (1933) [2] Davenport H., Sitzungsber. Preuss. Akad. Wiss. pp 830– (1933) [3] Del M., J. Théor. Nombres Bordeaux 6 (2) pp 327– (1994) · Zbl 0839.11041 · doi:10.5802/jtnb.118 [4] Elliott P. D. T. A., Probabilistic number theory, I: Mean-value theorems (1979) · Zbl 0431.10029 · doi:10.1007/978-1-4612-9989-9 [5] Glaisher W. L., Quaterly Journal of Math. 25 pp 347– (1891) [6] Martinet J., Séminaire de Théorie des Nombres (1973) [7] Tenenbaum G., Introduction à la théorie analytique et probabiliste des nombres,, 2. ed. (1995) · Zbl 0880.11001 [8] Wall C. R., The theory of arithmetic functions (Kalamazoo, Mich., 1971) pp 283– (1972) · doi:10.1007/BFb0058798
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.