# Divisor function 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 10 External links 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 {displaystyle sigma _{z}(n)=sum _{dmid n}d^{z},!,} where {displaystyle {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 (OEIS: A000005). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) (OEIS: A000203).

The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS: 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: {displaystyle {begin{aligned}sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\&=1+1+1+1+1+1=6,end{aligned}}} while σ1(12) is the sum of all the divisors: {displaystyle {begin{aligned}sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\&=1+2+3+4+6+12=28,end{aligned}}} and the aliquot sum s(12) of proper divisors is: {displaystyle {begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\&=1+2+3+4+6=16.end{aligned}}} Table of values The cases x = 2 to 5 are listed in OEIS: A001157 − OEIS: A001160, x = 6 to 24 are listed in OEIS: A013954 − OEIS: A013972.

n factorization 0(n) 1(n) 2(n) 3(n) 4(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, {displaystyle {begin{aligned}sigma _{0}(p)&=2\sigma _{0}(p^{n})&=n+1\sigma _{1}(p)&=p+1end{aligned}}} because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the primorial, {displaystyle sigma _{0}(p_{n}#)=2^{n}} since n prime factors allow a sequence of binary selection ( {displaystyle p_{i}} or 1) from n terms for each proper divisor formed.

Clearly, {displaystyle 12} , and {displaystyle sigma _{x}(n)>n} for all {displaystyle n>1} , {displaystyle x>0} .

The divisor function is multiplicative (since each divisor c of the product mn with {displaystyle gcd(m,n)=1} distinctively correspond to a divisor a of m and a divisor b of n), but not completely multiplicative: {displaystyle gcd(a,b)=1Longrightarrow sigma _{x}(ab)=sigma _{x}(a)sigma _{x}(b).} The consequence of this is that, if we write {displaystyle n=prod _{i=1}^{r}p_{i}^{a_{i}}} where r = ω(n) is the number of distinct prime factors of n, pi is the ith prime factor, and ai is the maximum power of pi by which n is divisible, then we have:  {displaystyle sigma _{x}(n)=prod _{i=1}^{r}sum _{j=0}^{a_{i}}p_{i}^{jx}=prod _{i=1}^{r}left(1+p_{i}^{x}+p_{i}^{2x}+cdots +p_{i}^{a_{i}x}right).} which, when x ≠ 0, is equivalent to the useful formula:  {displaystyle sigma _{x}(n)=prod _{i=1}^{r}{frac {p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}}.} When x = 0, d(n) is:  {displaystyle sigma _{0}(n)=prod _{i=1}^{r}(a_{i}+1).} This result can be directly deduced from the fact that all divisors of {displaystyle n} are uniquely determined by the distinct tuples {displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})} of integers with {displaystyle 0leq x_{i}leq a_{i}} (i.e. {displaystyle a_{i}+1} independent choices for each {displaystyle x_{i}} ).

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 {displaystyle sigma _{0}(24)} as so: {displaystyle sigma _{0}(24)=prod _{i=1}^{2}(a_{i}+1)=(3+1)(1+1)=4cdot 2=8.} 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: {displaystyle {begin{aligned}sigma (n)&=sigma (n-1)+sigma (n-2)-sigma (n-5)-sigma (n-7)+sigma (n-12)+sigma (n-15)+cdots \[12mu]&=sum _{iin mathbb {N} }(-1)^{i+1}left(sigma left(n-{frac {1}{2}}left(3i^{2}-iright)right)+sigma left(n-{frac {1}{2}}left(3i^{2}+iright)right)right)end{aligned}}} where {displaystyle sigma (0)=n} if it occurs and {displaystyle sigma (x)=0} for {displaystyle x<0} , and {displaystyle {tfrac {1}{2}}left(3i^{2}mp iright)} are consecutive pairs of generalized pentagonal numbers (OEIS: A001318, starting at offset 1). 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 {displaystyle sigma _{0}(n)} is even; for a square integer, one divisor (namely {displaystyle {sqrt {n}}} ) is not paired with a distinct divisor and {displaystyle sigma _{0}(n)} is odd. Similarly, the number {displaystyle sigma _{1}(n)} is odd if and only if n is a square or twice a square. 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, {displaystyle n=2^{k}} , then {displaystyle sigma (n)=2cdot 2^{k}-1=2n-1} and {displaystyle s(n)=n-1} , which makes n almost-perfect. As an example, for two primes {displaystyle p,q:p1,s>a+1,} which for d(n) = σ0(n) gives:  {displaystyle sum _{n=1}^{infty }{frac {d(n)}{n^{s}}}=zeta ^{2}(s)quad {text{for}}quad s>1,} and a Ramanujan identity {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)}},} which is a special case of the Rankin–Selberg convolution.

A Lambert series involving the divisor function is:  {displaystyle sum _{n=1}^{infty }q^{n}sigma _{a}(n)=sum _{n=1}^{infty }sum _{j=1}^{infty }n^{a}q^{j,n}=sum _{n=1}^{infty }{frac {n^{a}q^{n}}{1-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.

For {displaystyle k>0} , there is an explicit series representation with Ramanujan sums {displaystyle c_{m}(n)} as : {displaystyle sigma _{k}(n)=zeta (k+1)n^{k}sum _{m=1}^{infty }{frac {c_{m}(n)}{m^{k+1}}}.} The computation of the first terms of {displaystyle c_{m}(n)} shows its oscillations around the "average value" {displaystyle zeta (k+1)n^{k}} : {displaystyle sigma _{k}(n)=zeta (k+1)n^{k}left[1+{frac {(-1)^{n}}{2^{k+1}}}+{frac {2cos {frac {2pi n}{3}}}{3^{k+1}}}+{frac {2cos {frac {pi n}{2}}}{4^{k+1}}}+cdots right]} Growth rate In little-o notation, the divisor function satisfies the inequality: {displaystyle {mbox{for all }}varepsilon >0,quad d(n)=o(n^{varepsilon }).} More precisely, Severin Wigert showed that: {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, {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: {displaystyle {mbox{for all }}xgeq 1,sum _{nleq x}d(n)=xlog x+(2gamma -1)x+O({sqrt {x}}),} where {displaystyle gamma } is Euler's gamma constant. Improving the bound {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:  {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: {displaystyle lim _{nto infty }{frac {1}{log n}}prod _{pleq n}{frac {p}{p-1}}=e^{gamma },} where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality: {displaystyle sigma (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: {displaystyle sigma (n) 1, where {displaystyle H_{n}} 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 ^ Jump up to: a b Long (1972, p. 46) ^ Pettofrezzo & Byrkit (1970, p. 63) ^ Pettofrezzo & Byrkit (1970, p. 58) ^ Jump up to: a b c Hardy & Wright (2008), pp. 310 f, §16.7. ^ Euler, Leonhard; Bell, Jordan (2004). "An observation on the sums of divisors". 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). ^ Jump up to: 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. ^ Jump up to: 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. 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 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, arXiv:1110.5078, Bibcode:2011arXiv1110.5078C 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, S2CID 3207238, Zbl 1163.11059 Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", The American Mathematical Monthly, 74: 969–973, doi:10.2307/2315280, JSTOR 2315280, MR 0220659 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 Hardy, G. H.; Wright, E. M. (2008) , An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, 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, 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, S2CID 15884740 Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950 Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77081766 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, S2CID 115619659 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, vol. 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, 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. show vte Divisibility-based sets of integers Categories: Divisor functionAnalytic number theoryNumber theoryZeta and L-functions

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