Divider count function
The number of divisors function indicates how many divisors a natural number has; the one and the number itself are counted. The number function is part of the mathematical branch of number theory . It is usually referred to as or - since it represents a special case of the divider function , also as .
Factorization of |
||
---|---|---|
1 | 1 | 1 |
2 | 2 | 2 |
3 | 4th | 2 2 |
4th | 6th | 2 · 3 |
5 | 16 | 2 4 |
6th | 12 | 2 2 3 |
7th | 64 | 2 6 |
8th | 24 | 2 3 3 |
9 | 36 | 2 2 3 2 |
10 | 48 | 2 4 3 |
11 | 1,024 | 2 10 |
12 | 60 | 2 2 3 5 |
13 | 4,096 | 2 12 |
14th | 192 | 2 6 3 |
15th | 144 | 2 4 3 2 |
16 | 120 | 2 3 3 5 |
17th | 65,536 | 2 16 |
18th | 180 | 2 2 3 2 5 |
19th | 262.144 | 2 18 |
20th | 240 | 2 4 3 5 |
21st | 576 | 2 6 3 2 |
22nd | 3,072 | 2 10 3 |
23 | 4,194,304 | 2 22 |
24 | 360 | 2 3 3 2 5 |
25th | 1,296 | 2 4 3 4 |
26th | 12,288 | 2 12 3 |
27 | 900 | 2 2 3 2 5 2 |
28 | 960 | 2 6 3 5 |
29 | 268.435.456 | 2 28 |
30th | 720 | 2 4 3 2 5 |
31 | 1,073,741,824 | 2 30 |
32 | 840 | 2 3 3 5 7 |
33 | 9.216 | 2 10 3 2 |
34 | 196.608 | 2 16 3 |
35 | 5,184 | 2 6 3 4 |
36 | 1,260 | 2 2 3 2 5 7 |
definition
For every natural number is defined:
- .
The first values are:
1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Divisor of | 1 | 1, 2 | 1, 3 | 1, 2, 4 | 1, 5 | 1, 2, 3, 6 | 1, 7 | 1, 2, 4, 8 | 1, 3, 9 | 1, 2, 5, 10 | 1, 11 | 1, 2, 3, 4, 6, 12 |
1 | 2 | 2 | 3 | 2 | 4th | 2 | 4th | 3 | 4th | 2 | 6th |
properties
- Does the number have the prime factorization
- so:
- For coprime numbers and the following applies:
- The number of divisors function is therefore a multiplicative number theoretic function .
- A number is a prime number if and only if holds.
- A number is a square number if and only if is odd.
- The Dirichlet series belonging to the number function is the square of the Riemann zeta function :
- (for ).
Asymptotics
On average , more precisely: There are constants , so
applies. (" " Is a Landau symbol and the Euler-Mascheroni constant .)
The knowledge that a number is a divisor of roughly numbers can serve as a heuristic , so that the sum on the left is roughly too
(For the last step, see harmonic series .)
The value has already been proven by PGL Dirichlet ; the search for better values is therefore also known as the Dirichletian divisor problem .
Better values were given by GF Woronoi (1903, ), J. van der Corput (1922, ) and MN Huxley ( ). On the other hand, GH Hardy and E. Landau showed that must apply. The possible values for are still the subject of research.
Generalizations
The divisor function assigns the sum of the -th powers of its divisors to each number :
The divisor sum is the special case of the divider function for , and the divisor number function is the special case of the divider function for :
See also
literature
- GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 .
Web links
- Eric W. Weisstein : Divisor Function . In: MathWorld (English).
swell
- ↑ For further initial values see also sequence A000005 in OEIS .
- ^ GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , Theorem 273, p. 239.
- ^ GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , Theorem 289, p. 250.
- ^ GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , Theorem 320, p. 264.
- ↑ PGL Dirichlet: About the determination of the mean values in number theory. In: Treatises of the Royal Prussian Academy of Sciences. 1849, pp. 69-83; or Werke, Volume II, pp. 49-66.
- ↑ G. Voronoï : Sur un problème du calcul des fonctions asymptotiques. In: J. Reine Angew. Math. 126 (1903) pp. 241-282.
- ↑ JG van der Corput: tightening of the estimation in the divider problem. In: Math. Ann. 87 (1922) 39-65. Corrections 89 (1923) p. 160.
- ^ MN Huxley: Exponential Sums and Lattice Points III . In: Proc. London Math. Soc. tape 87 , no. 3 , 2003, p. 591-609 .
-
^ GH Hardy: On Dirichlet's divisor problem. In: Lond. MS Proc. (2) 15 (1915) 1-25.
See GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , p. 272. - ↑ Eric W. Weisstein : Divisor Function . In: MathWorld (English).