×

Estimating the density of the abundant numbers. (English) Zbl 1452.11008

A positive integer \(n\) is said to be abundant if the sum of proper divisors of \(n\) exceeds \(n\). H. Davenport [Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, 830–837 (1933; Zbl 0008.19701)] showed that the set of abundant numbers has a natural density \(\mathbf d\). Several authors gave estimates for the value of \(\mathbf d\), and the best current estimate determining \(\mathbf d=0.2476\dots\) to four decimal places is due to M. Kobayashi [Int. J. Number Theory 10, No. 1, 73–84 (2014; Zbl 1288.11094)].
The paper under review gives detailed examination of the number of abundant numbers in intervals of \(10^6\) consecutive integers. The authors give an unhurried discussion of this topic replete with extensive numerical calculations and accompanying plots. It is argued that this evidence points to the fact that \(\mathbf{d}=0.24761\dots\) to five decimal places.
In addition to numerics, the authors show that any interval of \(10^6\) consecutive integers contain at least 237111 numbers which are either abundant or perfect.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N60 Distribution functions associated with additive and positive multiplicative functions
PDFBibTeX XMLCite
Full Text: Link

References:

[1] F. Behrend, ¨Uber numeri abundantes, Sitzungsber. Akad. Berlin (1932), 322-328. · Zbl 0005.24601
[2] F. Behrend, ¨Uber numeri abundantes, ii, Sitzungsber. Akad. Berlin (1933), 280-293. · Zbl 0006.39601
[3] H. Davenport, ¨Uber numeri abundantes, Sitzungsber. Akad. Berlin (1933), 830-837. · JFM 59.0948.04
[4] M. Del´eglise, Bounds for the density of abundant integers, Exp. Math. 7, no. 2 (1998), 137-143. · Zbl 0923.11127
[5] P. Erd˝os, Note on consecutive abundant numbers, J. Lond. Math. Soc. S1-10, no. 1 (1935), 128. · JFM 61.0130.03
[6] M. Kobayashi, On the Density of Abundant Numbers, Ph.D. thesis, Dartmouth College, 2010.
[7] M. Kobayashi, A new series for the density of abundant numbers, Int. J. Number Theory 10, no. 1 (2014), 73-84. · Zbl 1288.11094
[8] H R. K¨unsch, The jackknife and the bootstrap for general stationary observations, Ann. Statist. 17 no. 3 (1989), 1217-1241. · Zbl 0684.62035
[9] J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Math. Acad. Sci. Paris 158 (1914), 1869-1872. · JFM 45.0305.01
[10] I. Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc. 57, no. 6 (1951), 420-434. · Zbl 0044.03603
[11] P. Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, American Mathematical Society, Providence, RI, 2009. · Zbl 1187.11001
[12] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. · Zbl 0122.05001
[13] Hans Sali´e, ¨Uber die Dichte abundanter Zahlen, Math. Nachr. 14, no. 1 (1955), 39-46. · Zbl 0066.03005
[14] C. R Wall, P. L. Crews, and D. B. Johnson, Density bounds for the sum of divisors function, Math. Comp. 26, no. 119 (1972), 773-777. · Zbl 0258.10025
[15] C. R. Wall, Topics Related to the Sum of Unitary Divisors of an Integer, Ph.D. thesis, University of Tennessee, 1970.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.