Divisor function
In
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
Definition
The sum of positive divisors function σz(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as
where is shorthand for "d
The
Example
For example, σ0(12) is the number of the divisors of 12:
while σ1(12) is the sum of all the divisors:
and the aliquot sum s(12) of proper divisors is:
σ-1(n) is sometimes called the
Table of values
The cases x = 2 to 5 are listed in OEIS: A001157 through OEIS: A001160, x = 6 to 24 are listed in OEIS: A013954 through OEIS: A013972.
n | factorization | 𝜎0(n) | 𝜎1(n) | 𝜎2(n) | 𝜎3(n) | 𝜎4(n) |
---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 2 | 2 | 3 | 5 | 9 | 17 |
3 | 3 | 2 | 4 | 10 | 28 | 82 |
4 | 22 | 3 | 7 | 21 | 73 | 273 |
5 | 5 | 2 | 6 | 26 | 126 | 626 |
6 | 2×3 | 4 | 12 | 50 | 252 | 1394 |
7 | 7 | 2 | 8 | 50 | 344 | 2402 |
8 | 23 | 4 | 15 | 85 | 585 | 4369 |
9 | 32 | 3 | 13 | 91 | 757 | 6643 |
10 | 2×5 | 4 | 18 | 130 | 1134 | 10642 |
11 | 11 | 2 | 12 | 122 | 1332 | 14642 |
12 | 22×3 | 6 | 28 | 210 | 2044 | 22386 |
13 | 13 | 2 | 14 | 170 | 2198 | 28562 |
14 | 2×7 | 4 | 24 | 250 | 3096 | 40834 |
15 | 3×5 | 4 | 24 | 260 | 3528 | 51332 |
16 | 24 | 5 | 31 | 341 | 4681 | 69905 |
17 | 17 | 2 | 18 | 290 | 4914 | 83522 |
18 | 2×32 | 6 | 39 | 455 | 6813 | 112931 |
19 | 19 | 2 | 20 | 362 | 6860 | 130322 |
20 | 22×5 | 6 | 42 | 546 | 9198 | 170898 |
21 | 3×7 | 4 | 32 | 500 | 9632 | 196964 |
22 | 2×11 | 4 | 36 | 610 | 11988 | 248914 |
23 | 23 | 2 | 24 | 530 | 12168 | 279842 |
24 | 23×3 | 8 | 60 | 850 | 16380 | 358258 |
25 | 52 | 3 | 31 | 651 | 15751 | 391251 |
26 | 2×13 | 4 | 42 | 850 | 19782 | 485554 |
27 | 33 | 4 | 40 | 820 | 20440 | 538084 |
28 | 22×7 | 6 | 56 | 1050 | 25112 | 655746 |
29 | 29 | 2 | 30 | 842 | 24390 | 707282 |
30 | 2×3×5 | 8 | 72 | 1300 | 31752 | 872644 |
31 | 31 | 2 | 32 | 962 | 29792 | 923522 |
32 | 25 | 6 | 63 | 1365 | 37449 | 1118481 |
33 | 3×11 | 4 | 48 | 1220 | 37296 | 1200644 |
34 | 2×17 | 4 | 54 | 1450 | 44226 | 1419874 |
35 | 5×7 | 4 | 48 | 1300 | 43344 | 1503652 |
36 | 22×32 | 9 | 91 | 1911 | 55261 | 1813539 |
37 | 37 | 2 | 38 | 1370 | 50654 | 1874162 |
38 | 2×19 | 4 | 60 | 1810 | 61740 | 2215474 |
39 | 3×13 | 4 | 56 | 1700 | 61544 | 2342084 |
40 | 23×5 | 8 | 90 | 2210 | 73710 | 2734994 |
41 | 41 | 2 | 42 | 1682 | 68922 | 2825762 |
42 | 2×3×7 | 8 | 96 | 2500 | 86688 | 3348388 |
43 | 43 | 2 | 44 | 1850 | 79508 | 3418802 |
44 | 22×11 | 6 | 84 | 2562 | 97236 | 3997266 |
45 | 32×5 | 6 | 78 | 2366 | 95382 | 4158518 |
46 | 2×23 | 4 | 72 | 2650 | 109512 | 4757314 |
47 | 47 | 2 | 48 | 2210 | 103824 | 4879682 |
48 | 24×3 | 10 | 124 | 3410 | 131068 | 5732210 |
49 | 72 | 3 | 57 | 2451 | 117993 | 5767203 |
50 | 2×52 | 6 | 93 | 3255 | 141759 | 6651267 |
Properties
Formulas at prime powers
For a prime number p,
because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the primorial,
since n prime factors allow a sequence of binary selection ( or 1) from n terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a power of two; instead, the smallest such number may be obtained by multiplying together the first n Fermi–Dirac primes, prime powers whose exponent is a power of two.[4]
Clearly, for all , and for all , .
The divisor function is multiplicative (since each divisor c of the product mn with distinctively correspond to a divisor a of m and a divisor b of n), but not completely multiplicative:
The consequence of this is that, if we write
where r = ω(n) is the
which, when x ≠ 0, is equivalent to the useful formula: [5]
When x = 0, is: [5]
This result can be directly deduced from the fact that all divisors of are uniquely determined by the distinct tuples of integers with (i.e. independent choices for each ).
For example, if n is 24, there are two prime factors (p1 is 2; p2 is 3); noting that 24 is the product of 23×31, a1 is 3 and a2 is 1. Thus we can calculate as so:
The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
Other properties and identities
where if it occurs and for , and are consecutive pairs of generalized
For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and is odd. Similarly, the number is odd if and only if n is a square or twice a square.[9]
We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, that is, the divisors of n excluding n itself. This function is used to recognize perfect numbers, which are the n such that s(n) = n. If s(n) > n, then n is an abundant number, and if s(n) < n, then n is a deficient number.
If n is a power of 2, , then and , which makes n almost-perfect.
As an example, for two primes , let
- .
Then
and
where is
Then, the roots of
express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n or , as
Also, knowing n and either or , or, alternatively, and either or allows an easy recovery of p and q.
In 1984, Roger Heath-Brown proved that the equality
is true for infinitely many values of n, see OEIS: A005237.
Series relations
Two Dirichlet series involving the divisor function are: [10]
where is the Riemann zeta function. The series for d(n) = σ0(n) gives: [10]
and a
which is a special case of the Rankin–Selberg convolution.
A Lambert series involving the divisor function is: [12]
for arbitrary
For , there is an explicit series representation with
The computation of the first terms of shows its oscillations around the "average value" :
Growth rate
In little-o notation, the divisor function satisfies the inequality:[14][15]
More precisely,
On the other hand, since there are infinitely many prime numbers,[15]
In
where is
The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: [18]
where lim sup is the
where p denotes a prime.
In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, Robin's inequality
- (where γ is the Euler–Mascheroni constant)
holds for all sufficiently large n (Ramanujan 1997). The largest known value that violates the inequality is n=5040. In 1984, Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n > 5040 must be superabundant (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime (Choie et al. 2007).
Robin also proved, unconditionally, that the inequality:
holds for all n ≥ 3.
A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:
for every natural number n > 1, where is the nth harmonic number, (Lagarias 2002).
See also
- Divisor sum convolutions, lists a few identities involving the divisor functions
- Euler's totient function, Euler's phi function
- Refactorable number
- Table of divisors
- Unitary divisor
Notes
- ^ a b Long (1972, p. 46)
- ^ Pettofrezzo & Byrkit (1970, p. 63)
- ^ Pettofrezzo & Byrkit (1970, p. 58)
- doi:10.1112/plms/s2_14.1.347; see section 47, pp. 405–406, reproduced in Collected Papers of Srinivasa Ramanujan, Cambridge Univ. Press, 2015, pp. 124–125
- ^ a b c Hardy & Wright (2008), pp. 310 f, §16.7.
- arXiv:math/0411587.
- ^ https://scholarlycommons.pacific.edu/euler-works/175/, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs
- ^ https://scholarlycommons.pacific.edu/euler-works/542/, De mirabilis proprietatibus numerorum pentagonalium
- ^ Gioia & Vaidya (1967).
- ^ a b Hardy & Wright (2008), pp. 326–328, §17.5.
- ^ Hardy & Wright (2008), pp. 334–337, §17.8.
- ^ Hardy & Wright (2008), pp. 338–341, §17.10.
- ^ E. Krätzel (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130. (German)
- ^ Apostol (1976), p. 296.
- ^ a b c Hardy & Wright (2008), pp. 342–347, §18.1.
- ^ Apostol (1976), Theorem 3.3.
- ^ Hardy & Wright (2008), pp. 347–350, §18.2.
- ^ Hardy & Wright (2008), pp. 469–471, §22.9.
References
- Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis" (PDF), American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128, archived from the original(PDF) on 2014-04-11.
- Zbl 0335.10001
- ISBN 0-262-02405-5, see page 234 in section 8.8.
- Caveney, Geoffrey; Bibcode:2011arXiv1110.5078C
- Zbl 1163.11059
- Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", MR 0220659
- Zbl 1159.11001
- Ivić, Aleksandar (1985), The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York etc.: John Wiley & Sons, pp. 385–440, Zbl 0556.10026
- S2CID 15884740
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: LCCN 77171950
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: LCCN 77081766
- S2CID 115619659
- Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", MR 0774171
- Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, vol. 76, Cambridge: Zbl 1227.11002
External links
- Weisstein, Eric W. "Divisor Function". MathWorld.
- Weisstein, Eric W. "Robin's Theorem". MathWorld.
- Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.