| 释义 |
Divisor Function is defined as the sum of the  th Powers of the Divisors of  . Thefunction  gives the total number of Divisors of and is often denoted  ,  , , or  (Hardy and Wright 1979, pp. 354-355). The first few values of are 1, 2, 2, 3, 2, 4,2, 4, 3, 4, 2, 6, ... (Sloane's A000005). The function  is equal to the sum of Divisors of and is often denoted  . The first few values of are 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, ...(Sloane's A000203). The first few values of  are 1, 5, 10, 21, 26, 50, 50, 85, 91, 130, ... (Sloane's A001157). Thefirst few values of  are 1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, ... (Sloane's A001158).
The sum of the Divisors of excluding itself (i.e., the Proper Divisors of ) is called the Restricted Divisor Function and is denoted  . The first few values are 0, 1,1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (Sloane's A001065).
As an illustrative example, consider the number 140, which has Divisors  , 2, 4, 5, 7, 10, 14, 20,28, 35, 70, and 140 (for a total of  of them). Therefore,
The function has the series expansion  | |  | (5) |
(Hardy 1959). It also satisfies the Inequality
 | (6) |
 is the Euler-Mascheroni Constant (Robin 1984, Erdös 1989).
Let a number have Prime factorization
 | (7) |
 | (8) |
 | (9) |
In general,
 | (10) |
 showed that the average number of Divisors of all numbers from 1 to isasymptotic to
 | (11) |
A curious identity derived using Modular Form theory is given by
 | (12) |
The asymptotic Summatory Function of  is given by
 | (13) |
 | (14) |
 with  are
 | (15) |
 ,
 | (16) |
 | (17) |
 for Prime are given by Sorli.See also Dirichlet Divisor Problem, Divisor, Factor, Greatest Prime Factor, Gronwall'sTheorem, Least Prime Factor, Multiply Perfect Number, Ore's Conjecture, Perfect Number,r(n), Restricted Divisor Function, Silverman Constant, Tau Function, Totient Function,Totient Valence Function, Twin Peaks
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Divisor Functions.'' §24.3.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 827, 1972.Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 279-325, 1952. Dirichlet, G. L. ``Sur l'usage des séries infinies dans la théorie des nombres.'' J. reine angew. Math. 18, 259-274, 1838. Erdös, P. ``Ramanujan and I.'' In Proceedings of the International Ramanujan Centenary Conference held at Anna University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer-Verlag, pp. 1-20, 1989. Guy, R. K. ``Solutions of  ,'' ``Analogs with , ,'' ``Solutions of  ,'' and ``Solutions of  .'' §B11, B12, B13 and B15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 67-70, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 141, 1959. Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 354-355, 1979. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 250-251, 1991. Robin, G. ``Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann.'' J. Math. Pures Appl. 63, 187-213, 1984. Sloane, N. J. A. SequencesA000005/M0246,A000203/M2329A001065/M2226,A001157/M3799, andA001158/M4605,in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Subbarao, M. V. ``On Two Congruences for Primality.'' Pacific J. Math. 52, 261-268, 1974.
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