Divisor function

Divisor function (Redirected from Robin's theorem) Vai alla navigazione Vai alla ricerca "Robin's theorem" reindirizza qui. For Robbins' theorem in graph theory, see Robbins' theorem. 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 matematica, 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 (Compreso 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, quale, as the name implies, is a sum over the divisor function.

Contenuti 1 Definizione 2 Esempio 3 Table of values 4 Proprietà 4.1 Formulas at prime powers 4.2 Other properties and identities 5 Series relations 6 Growth rate 7 Guarda anche 8 Appunti 9 Riferimenti 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)=somma _{dmid n}d^{z},!,} dove {stile di visualizzazione {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] (OEIS: A000005). When z 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) (OEIS: A000203).

The aliquot sum s(n) of n is the sum of the proper divisors (questo è, 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: {stile di visualizzazione {inizio{allineato}sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\&=1+1+1+1+1+1=6,end{allineato}}} while σ1(12) is the sum of all the divisors: {stile di visualizzazione {inizio{allineato}sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\&=1+2+3+4+6+12=28,end{allineato}}} and the aliquot sum s(12) of proper divisors is: {stile di visualizzazione {inizio{allineato}S(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\&=1+2+3+4+6=16.end{allineato}}} Table of values The cases x = 2 a 5 are listed in OEIS: A001157 − OEIS: A001160, x = 6 a 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, {stile di visualizzazione {inizio{allineato}sigma _{0}(p)&=2\sigma _{0}(p^{n})&=n+1\sigma _{1}(p)&=p+1end{allineato}}} because by definition, the factors of a prime number are 1 and itself. Anche, where pn# denotes the primorial, {displaystyle sigma _{0}(p_{n}#)=2^{n}} since n prime factors allow a sequence of binary selection ( {stile di visualizzazione p_{io}} o 1) from n terms for each proper divisor formed.

Chiaramente, {stile di visualizzazione 12} , e {displaystyle sigma _{X}(n)>n} per tutti {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(un,b)=1Longrightarrow sigma _{X}(ab)=sigma _{X}(un)sigma _{X}(b).} The consequence of this is that, se scriviamo {displaystyle n=prod _{io=1}^{r}p_{io}^{un_{io}}} 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, Poi abbiamo: [4] {displaystyle sigma _{X}(n)=prodotto _{io=1}^{r}somma _{j=0}^{un_{io}}p_{io}^{jx}=prodotto _{io=1}^{r}sinistra(1+p_{io}^{X}+p_{io}^{2X}+cdots +p_{io}^{un_{io}X}Giusto).} quale, when x ≠ 0, is equivalent to the useful formula: [4] {displaystyle sigma _{X}(n)=prodotto _{io=1}^{r}{frac {p_{io}^{(un_{io}+1)X}-1}{p_{io}^{X}-1}}.} When x = 0, d(n) è: [4] {displaystyle sigma _{0}(n)=prodotto _{io=1}^{r}(un_{io}+1).} This result can be directly deduced from the fact that all divisors of {stile di visualizzazione n} are uniquely determined by the distinct tuples {stile di visualizzazione (X_{1},X_{2},...,X_{io},...,X_{r})} of integers with {displaystyle 0leq x_{io}leq a_{io}} (cioè. {stile di visualizzazione a_{io}+1} independent choices for each {stile di visualizzazione x_{io}} ).

Per esempio, 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)=prodotto _{io=1}^{2}(un_{io}+1)=(3+1)(1+1)=4cdot 2=8.} The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, e 24.

Other properties and identities Euler proved the remarkable recurrence:[5][6][7] {stile di visualizzazione {inizio{allineato}sigma (n)&=sigma (n-1)+sigma (n-2)-sigma (n-5)-sigma (n-7)+sigma (n-12)+sigma (n-15)+cdot \[12in]&=sum _{iin mathbb {N} }(-1)^{io+1}sinistra(sigma a sinistra(n-{frac {1}{2}}sinistra(3i^{2}-iright)Giusto)+sigma a sinistra(n-{frac {1}{2}}sinistra(3i^{2}+iright)Giusto)Giusto)fine{allineato}}} dove {displaystyle sigma (0)=n} if it occurs and {displaystyle sigma (X)=0} per {stile di visualizzazione 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.[8] 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) dà: [9] {somma dello stile di visualizzazione _{n=1}^{infty }{frac {d(n)}{n^{S}}}=zeta ^{2}(S)quad {testo{per}}quad s>1,} and a Ramanujan identity[10] {somma dello stile di visualizzazione _{n=1}^{infty }{frac {sigma _{un}(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: [11] {somma dello stile di visualizzazione _{n=1}^{infty }q^{n}sigma _{un}(n)=somma _{n=1}^{infty }somma _{j=1}^{infty }n^{un}q^{j,n}=somma _{n=1}^{infty }{frac {n^{un}q^{n}}{1-q^{n}}}} for arbitrary complex |q| ≤ 1 e un. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.

Per {displaystyle k>0} , there is an explicit series representation with Ramanujan sums {stile di visualizzazione c_{m}(n)} come :[12] {displaystyle sigma _{K}(n)=zeta (k+1)n^{K}somma _{m=1}^{infty }{frac {c_{m}(n)}{m^{k+1}}}.} The computation of the first terms of {stile di visualizzazione 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}sinistra[1+{frac {(-1)^{n}}{2^{k+1}}}+{frac {2cos {frac {2spillo}{3}}}{3^{k+1}}}+{frac {2cos {frac {spillo}{2}}}{4^{k+1}}}+cdot a destra]} Growth rate In little-o notation, the divisor function satisfies the inequality:[13][14] {stile di visualizzazione {mbox{per tutti }}varepsilon >0,quad d(n)=o(n^{varepsilon }).} Più precisamente, Severin Wigert showed that:[14] {displaystyle limsup _{infty }{frac {log d(n)}{log n/log log n}}= registro 2.} D'altro canto, since there are infinitely many prime numbers,[14] {displaystyle liminf _{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:[15][16] {stile di visualizzazione {mbox{per tutti }}xgeq 1,sum _{nleq x}d(n)=xlog x+(2gamma -1)x+O({mq {X}}),} dove {gamma di stili di visualizzazione } is Euler's gamma constant. Improving the bound {stile di visualizzazione O({mq {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: [17] {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, pubblicato in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that: {displaystyle lim _{infty }{frac {1}{registro n}}pungolo _{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, dove {stile di visualizzazione 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) ^ Salta su: a b c Hardy & Wright (2008), pp. 310 f, §16.7. ^ Euler, Leonhard; Campana, 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). ^ Salta su: 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. Berlino: VEB Deutscher Verlag der Wissenschaften. p. 130. (Tedesco) ^ Apostolo (1976), p. 296. ^ Salta su: a b c Hardy & Wright (2008), pp. 342–347, §18.1. ^ Apostolo (1976), Teorema 3.3. ^ Hardy & Wright (2008), pp. 347–350, §18.2. ^ Hardy & Wright (2008), pp. 469–471, §22.9. References Akbary, Emiro; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis" (PDF), Mensile matematico americano, 116 (3): 273–275, doi:10.4169/193009709X470128, archiviato dall'originale (PDF) Su 2014-04-11. Apostolo, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, SIG 0434929, Zbl 0335.10001 Bach, Eric; Shallit, Jeffrey, Algorithmic Number Theory, volume 1, 1996, MIT stampa. ISBN 0-262-02405-5, see page 234 in section 8.8. Caveney, Geoffrey; Nicola, Jean Louis; Sondow, Jonathan (2011), "Robin's theorem, primi, 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, Nicola; Moree, Pieter; Solé, Patrizio (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, SIG 2394891, S2CID 3207238, Zbl 1163.11059 Gioia, UN. UN.; Vaidya, UN. M. (1967), "Amicable numbers with opposite parity", Il mensile matematico americano, 74: 969–973, doi:10.2307/2315280, JSTOR 2315280, SIG 0220659 Grönwall, Thomas Hakon (1913), "Some asymptotic expressions in the theory of numbers", Transazioni dell'American Mathematical Society, 14: 113–122, doi:10.1090/S0002-9947-1913-1500940-6 Hardy, G. H.; Wright, e. M. (2008) [1938], Un'introduzione alla teoria dei numeri, Revised by D. R. Heath-Brown and J. H. Argentiere. Foreword by Andrew Wiles. (6th ed.), Oxford: la stampa dell'università di Oxford, ISBN 978-0-19-921986-5, SIG 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", Il mensile matematico americano, 109 (6): 534–543, arXiv:math/0008177, doi:10.2307/2695443, ISSN 0002-9890, JSTOR 2695443, SIG 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: Sala dell'Apprendista, 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/UN:1009764017495, ISSN 1382-4090, SIG 1606180, S2CID 115619659 Robin, Tipo (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, SIG 0774171 Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, Testi degli studenti della London Mathematical Society, vol. 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002 Weissstein esterno sinistro, Eric W. "Divisor Function". Math World. Weisstein, Eric W. "Robin's Theorem". Math World. Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (cioè. 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