Divisor function

"Robin's theorem" redirects here. For Robbins' theorem in graph theory, see Robbins' theorem.

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (January 2016) (Learn how and when to remove this template message)

Divisor function σ₀(n) up to n = 250

Sigma function σ₁(n) up to n = 250

Sum of the squares of divisors, σ₂(n), up to n = 250

Sum of cubes of divisors, σ₃(n) up to n = 250

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Contents

• 
  1 Definition
  


• 
  2 Example
  


• 
  3 Table of values
  


• 
  4 Properties
  
  • 
    4.1 Formulas at prime powers
    
  
  
  • 
    4.2 Other properties and identities
    
  
  


• 
  5 Series relations
  


• 
  6 Growth rate
  


• 
  7 See also
  


• 
  8 Notes
  


• 
  9 References
  

Definition

The sum of positive divisors function σ_x(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as

σx(n)=∑d∣ndx,displaystyle sigma _x(n)=sum _dmid nd^x,!,

where d∣ndisplaystyle dmid n is shorthand for "d divides n".
The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ₀(n), or the number-of-divisors function^[1][2] ( A000005). When x is 1, the function is called the sigma function or sum-of-divisors function,^[1][3] and the subscript is often omitted, so σ(n) is the same as σ₁(n) ( A000203).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself,  A001065), and equals σ₁(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.

Example

For example, σ₀(12) is the number of the divisors of 12:

σ0(12)=10+20+30+40+60+120=1+1+1+1+1+1=6,displaystyle beginalignedsigma _0(12)&=1^0+2^0+3^0+4^0+6^0+12^0\&=1+1+1+1+1+1=6,endaligned& = "1 + 1 + 1 + 1 + 1 + 1 = 6,"
endalign
"/>

while σ₁(12) is the sum of all the divisors:

σ1(12)=11+21+31+41+61+121=1+2+3+4+6+12=28,displaystyle beginalignedsigma _1(12)&=1^1+2^1+3^1+4^1+6^1+12^1\&=1+2+3+4+6+12=28,endaligned& = "1 + 2 + 3 + 4 + 6 + 12 = 28,"
endalign
"/>

and the aliquot sum s(12) of proper divisors is:

s(12)=11+21+31+41+61=1+2+3+4+6=16.displaystyle beginaligneds(12)&=1^1+2^1+3^1+4^1+6^1\&=1+2+3+4+6=16.endaligned& = "1 + 2 + 3 + 4 + 6 = 16."
endalign
"/>

Table of values

n	Divisors	σ0(n)	σ1(n)	s(n) = σ1(n) − n	Comment
1	1	1	1	0	square number: σ0(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)
2	1, 2	2	3	1	Prime: σ1(n) = 1 + n so s(n) = 1; power of 2: s(n) = n − 1 (almost-perfect)
3	1, 3	2	4	1	Prime: σ1(n) = 1 + n so s(n) = 1
4	1, 2, 4	3	7	3	square number: σ0(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)
5	1, 5	2	6	1	Prime: σ1(n) = 1 + n so s(n) = 1
6	1, 2, 3, 6	4	12	6	first perfect number: s(n) = n
7	1, 7	2	8	1	Prime: σ1(n) = 1 + n so s(n) = 1
8	1, 2, 4, 8	4	15	7	power of 2: s(n) = n − 1 (almost-perfect)
9	1, 3, 9	3	13	4	square number: σ0(n) is odd
10	1, 2, 5, 10	4	18	8
11	1, 11	2	12	1	Prime: σ1(n) = 1 + n so s(n) = 1
12	1, 2, 3, 4, 6, 12	6	28	16	first abundant number: s(n) > n
13	1, 13	2	14	1	Prime: σ1(n) = 1 + n so s(n) = 1
14	1, 2, 7, 14	4	24	10
15	1, 3, 5, 15	4	24	9
16	1, 2, 4, 8, 16	5	31	15	square number: σ0(n) is odd; power of 2: s(n) = n − 1 (almost-perfect)

σ0(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	2	2	3	2	4	2	4	3	4	2
12+	6	2	4	4	5	2	6	2	6	4	4	2
24+	8	3	4	4	6	2	8	2	6	4	4	4
36+	9	2	4	4	8	2	8	2	6	6	4	2
48+	10	3	6	4	6	2	8	4	8	4	4	2
60+	12	2	4	6	7	4	8	2	6	4	8	2
72+	12	2	4	6	6	4	8	2	10	5	4	2
84+	12	4	4	4	8	2	12	4	6	4	4	4
96+	12	2	6	6	9	2	8	2	8	8	4	2
108+	12	2	8	4	10	2	8	4	6	6	4	4
120+	16	3	4	4	6	4	12	2	8	4	8	2
132+	12	4	4	8	8	2	8	2	12	4	4	4

σ1(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	3	4	7	6	12	8	15	13	18	12
12+	28	14	24	24	31	18	39	20	42	32	36	24
24+	60	31	42	40	56	30	72	32	63	48	54	48
36+	91	38	60	56	90	42	96	44	84	78	72	48
48+	124	57	93	72	98	54	120	72	120	80	90	60
60+	168	62	96	104	127	84	144	68	126	96	144	72
72+	195	74	114	124	140	96	168	80	186	121	126	84
84+	224	108	132	120	180	90	234	112	168	128	144	120
96+	252	98	171	156	217	102	216	104	210	192	162	108
108+	280	110	216	152	248	114	240	144	210	182	180	144
120+	360	133	186	168	224	156	312	128	255	176	252	132
132+	336	160	204	240	270	138	288	140	336	192	216	168

σ2(n)	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9	+10	+11
0+		1	5	10	21	26	50	50	85	91	130	122
12+	210	170	250	260	341	290	455	362	546	500	610	530
24+	850	651	850	820	1050	842	1300	962	1365	1220	1450	1300
36+	1911	1370	1810	1700	2210	1682	2500	1850	2562	2366	2650	2210
48+	3410	2451	3255	2900	3570	2810	4100	3172	4250	3620	4210	3482
60+	5460	3722	4810	4550	5461	4420	6100	4490	6090	5300	6500	5042
72+	7735	5330	6850	6510	7602	6100	8500	6242	8866	7381	8410	6890
84+	10500	7540	9250	8420	10370	7922	11830	8500	11130	9620	11050	9412
96+	13650	9410	12255	11102	13671	10202	14500	10610	14450	13000	14050	11450
108+	17220	11882	15860	13700	17050	12770	18100	13780	17682	15470	17410	14500
120+	22100	14763	18610	16820	20202	16276	22750	16130	21845	18500	22100	17162
132+	25620	18100	22450	21320	24650	18770	26500	19322	27300	22100	25210	20740

The cases x = 2 to 5 are listed in  A001157 −  A001160, x = 6 to 24 are listed in  A013954 −  A013972.

Properties

Formulas at prime powers

For a prime number p,

σ0(p)=2σ0(pn)=n+1σ1(p)=p+1displaystyle beginalignedsigma _0(p)&=2\sigma _0(p^n)&=n+1\sigma _1(p)&=p+1endaligned

because by definition, the factors of a prime number are 1 and itself. Also, where p_n# denotes the primorial,

σ0(pn#)=2ndisplaystyle sigma _0(p_n#)=2^n,

since n prime factors allow a sequence of binary selection (pidisplaystyle p_i or 1) from n terms for each proper divisor formed.

Clearly, 1<σ0(n)<ndisplaystyle 1<sigma _0(n)<n and σ(n) > n for all n > 2.

The divisor function is multiplicative, but not completely multiplicative. The consequence of this is that, if we write

n=∏i=1rpiaidisplaystyle n=prod _i=1^rp_i^a_i

where r = ω(n) is the number of distinct prime factors of n, p_i is the ith prime factor, and a_i is the maximum power of p_i by which n is divisible, then we have

σx(n)=∏i=1r∑j=0aipijx=∏i=1r(1+pix+pi2x+⋯+piaix).displaystyle sigma _x(n)=prod _i=1^rsum _j=0^a_ip_i^jx=prod _i=1^r(1+p_i^x+p_i^2x+cdots +p_i^a_ix)."/>

which is equivalent to the useful formula:

σx(n)=∏i=1rpi(ai+1)x−1pix−1displaystyle sigma _x(n)=prod _i=1^rfrac p_i^(a_i+1)x-1p_i^x-1

It follows (by setting x = 0) that d(n) is:

σ0(n)=∏i=1r(ai+1).displaystyle sigma _0(n)=prod _i=1^r(a_i+1).

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 σ0(24)displaystyle sigma _0(24) as so:

σ0(24)=∏i=12(ai+1)=(3+1)(1+1)=4⋅2=8.displaystyle beginalignedsigma _0(24)&=prod _i=1^2(a_i+1)\&=(3+1)(1+1)=4cdot 2=8.endaligned

The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.

Other properties and identities

Euler proved the remarkable recurrence:
^[4][5]

σ(n)=σ(n−1)+σ(n−2)−σ(n−5)−σ(n−7)+σ(n−12)+σ(n−15)+⋯=∑i∈Z(−1)i+1(σ(n−12(3i2−i))+δ(n,12(3i2−i))n)displaystyle beginarrayrclsigma (n)&=&sigma (n-1)+sigma (n-2)-sigma (n-5)-sigma (n-7)+sigma (n-12)+sigma (n-15)+cdots \&=&sum _iin mathbb Z (-1)^i+1left(sigma (n-frac 12(3i^2-i))+delta (n,frac 12(3i^2-i)),nright)endarray

where we set σ(i)=0displaystyle sigma (i)=0 for i≤0displaystyle ileq 0,
we use the Kronecker delta δ(⋅,⋅)displaystyle delta (cdot ,cdot ), and 12(3i2−i)displaystyle tfrac 12(3i^2-i) are the pentagonal numbers. Indeed, Euler proved this by logarithmic differentiation of the identity in his Pentagonal number theorem.

For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ0(n)displaystyle sigma _0(n) is even; for a square integer, one divisor (namely ndisplaystyle sqrt n) is not paired with a distinct divisor and σ0(n)displaystyle sigma _0(n) is odd. Similarly, the number σ1(n)displaystyle sigma _1(n) is odd if and only if n is a square or twice a square.^{[citation needed]}

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 the one used to recognize perfect numbers which are the n for which 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, for example, n=2kdisplaystyle n=2^k, then σ(n)=2⋅2k−1=2n−1,displaystyle sigma (n)=2cdot 2^k-1=2n-1, and s(n) = n - 1, which makes n almost-perfect.

As an example, for two distinct primes p and q with p < q, let

n=pq.displaystyle n=pq.,

Then

σ(n)=(p+1)(q+1)=n+1+(p+q),displaystyle sigma (n)=(p+1)(q+1)=n+1+(p+q),,

φ(n)=(p−1)(q−1)=n+1−(p+q),displaystyle varphi (n)=(p-1)(q-1)=n+1-(p+q),,

and

n+1=(σ(n)+φ(n))/2,displaystyle n+1=(sigma (n)+varphi (n))/2,,

p+q=(σ(n)−φ(n))/2,displaystyle p+q=(sigma (n)-varphi (n))/2,,

where φ(n)displaystyle varphi (n) is Euler's totient function.

Then, the roots of:

(x−p)(x−q)=x2−(p+q)x+n=x2−[(σ(n)−φ(n))/2]x+[(σ(n)+φ(n))/2−1]=0displaystyle (x-p)(x-q)=x^2-(p+q)x+n=x^2-[(sigma (n)-varphi (n))/2]x+[(sigma (n)+varphi (n))/2-1]=0,

allow us to express p and q in terms of σ(n) and φ(n) only, without even knowing n or p+q, as:

p=(σ(n)−φ(n))/4−[(σ(n)−φ(n))/4]2−[(σ(n)+φ(n))/2−1],displaystyle p=(sigma (n)-varphi (n))/4-sqrt [(sigma (n)-varphi (n))/4]^2-[(sigma (n)+varphi (n))/2-1],,

q=(σ(n)−φ(n))/4+[(σ(n)−φ(n))/4]2−[(σ(n)+φ(n))/2−1].displaystyle q=(sigma (n)-varphi (n))/4+sqrt [(sigma (n)-varphi (n))/4]^2-[(sigma (n)+varphi (n))/2-1].,

Also, knowing n and either σ(n)displaystyle sigma (n) or φ(n)displaystyle varphi (n) (or knowing p+q and either σ(n)displaystyle sigma (n) or φ(n)displaystyle varphi (n)) allows us to easily find p and q.

In 1984, Roger Heath-Brown proved that the equality

σ0(n)=σ0(n+1)displaystyle sigma _0(n)=sigma _0(n+1)

is true for an infinity of values of n, see  A005237.

Series relations

Two Dirichlet series involving the divisor function are:

∑n=1∞σa(n)ns=ζ(s)ζ(s−a)fors>1,s>a+1,displaystyle sum _n=1^infty frac sigma _a(n)n^s=zeta (s)zeta (s-a)quad textforquad s>1,s>a+1,

which for d(n) = σ₀(n) gives

∑n=1∞d(n)ns=ζ2(s)fors>1,displaystyle sum _n=1^infty frac d(n)n^s=zeta ^2(s)quad textforquad s>1,

and

∑n=1∞σa(n)σb(n)ns=ζ(s)ζ(s−a)ζ(s−b)ζ(s−a−b)ζ(2s−a−b).displaystyle sum _n=1^infty frac sigma _a(n)sigma _b(n)n^s=frac zeta (s)zeta (s-a)zeta (s-b)zeta (s-a-b)zeta (2s-a-b).

A Lambert series involving the divisor function is:

∑n=1∞qnσa(n)=∑n=1∞∑j=1∞naqjn=∑n=1∞naqn1−qndisplaystyle sum _n=1^infty q^nsigma _a(n)=sum _n=1^infty sum _j=1^infty n^aq^j,n=sum _n=1^infty frac n^aq^n1-q^n

for arbitrary complex |q| ≤ 1 and a. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

Growth rate

In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book^[6])

for all ϵ>0,d(n)=o(nϵ).displaystyle mboxfor all epsilon >0,quad d(n)=o(n^epsilon ).

More precisely, Severin Wigert showed that

lim supn→∞log⁡d(n)log⁡n/log⁡log⁡n=log⁡2.displaystyle limsup _nto infty frac log d(n)log n/log log n=log 2.

On the other hand, since there are infinitely many prime numbers,

lim infn→∞d(n)=2.displaystyle liminf _nto infty d(n)=2.

In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book^[6])

for all x≥1,∑n≤xd(n)=xlog⁡x+(2γ−1)x+O(x),displaystyle mboxfor all xgeq 1,sum _nleq xd(n)=xlog x+(2gamma -1)x+O(sqrt x),

where γdisplaystyle gamma is Euler's gamma constant. Improving the bound O(x)displaystyle O(sqrt x) in this formula is known as Dirichlet's divisor problem.

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by:

lim supn→∞σ(n)nlog⁡log⁡n=eγ,displaystyle limsup _nrightarrow infty frac sigma (n)n,log log n=e^gamma ,"/>

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that

limn→∞1log⁡n∏p≤npp−1=eγ,displaystyle lim _nto infty frac 1log nprod _pleq nfrac pp-1=e^gamma ,

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:

 σ(n)<eγnlog⁡log⁡ndisplaystyle sigma (n)<e^gamma nlog log n (Robin's inequality)

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

 σ(n)<eγnlog⁡log⁡n+0.6483 nlog⁡log⁡ndisplaystyle sigma (n)<e^gamma nlog log n+frac 0.6483 nlog log n

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

σ(n)<Hn+ln⁡(Hn)eHndisplaystyle sigma (n)<H_n+ln(H_n)e^H_n

for every natural number n > 1, where Hndisplaystyle H_n is the nth harmonic number, (Lagarias 2002).

See also

• 
  Euler's totient function (Euler's phi function)


• Table of divisors


• 
  Divisor sum convolutions Lists a few identities involving the divisor functions


• Unitary divisor


• Refactorable number

Notes

1. 
   ^ ^ab Long (1972, p. 46)
   


2. 
   ^ Pettofrezzo & Byrkit (1970, p. 63)
   


3. 
   ^ Pettofrezzo & Byrkit (1970, p. 58)
   


4. 
   ^ https://arxiv.org/abs/math/0411587, An observation on the sums of divisors
   


5. 
   ^ http://eulerarchive.maa.org//pages/E175.html, Decouverte d'une loi tout extraordinaire des nombres par rapport a la somme de leurs diviseurs
   


6. 
   ^ ^ab Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 
   

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 .


• 
  Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 234 in section 8.8.


• 
  Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011), "Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis" (PDF), INTEGERS: the Electronic Journal of Combinatorial Number Theory, 11: A33 
  


• 
  Choie, YoungJu; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), "On Robin's criterion for the Riemann hypothesis", Journal de théorie des nombres de Bordeaux, 19 (2): 357–372, arXiv:math.NT/0604314 , doi:10.5802/jtnb.591, ISSN 1246-7405, MR 2394891, Zbl 1163.11059 
  


• 
  Grönwall, Thomas Hakon (1913), "Some asymptotic expressions in the theory of numbers", Transactions of the American Mathematical Society, 14: 113–122, doi:10.1090/S0002-9947-1913-1500940-6 
  


• 
  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, ISBN 0-471-80634-X, Zbl 0556.10026 
  


• 
  Lagarias, Jeffrey C. (2002), "An elementary problem equivalent to the Riemann hypothesis", The American Mathematical Monthly, 109 (6): 534–543, arXiv:math/0008177 , doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, MR 1908008 
  


• 
  Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950 
  


• 
  Ramanujan, Srinivasa (1997), "Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin", The Ramanujan Journal, 1 (2): 119–153, doi:10.1023/A:1009764017495, ISSN 1382-4090, MR 1606180 
  


• 
  Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77081766 
  


• 
  Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 63 (2): 187–213, ISSN 0021-7824, MR 0774171 
  


• 
  Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002 
  


• 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.

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

Clash Royale CLAN TAG#URR8PPP