 Divisor function σ 0( n) up to n=250
 Sigma function σ 1( n) up to n=250
 Sum of the squares of divisors, σ 2( n), up to n=250
 Sum of cubes of divisors, σ 3( n) up to n=250 In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred as the divisor function, it counts the number of divisors of an integer. 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. Image File history File links Divisor. ...
Image File history File links Divisor. ...
Image File history File links Sigma_function. ...
Image File history File links Sigma_function. ...
Image File history File links Divisor_square. ...
Image File history File links Divisor_square. ...
Image File history File links Divisor_cube. ...
Image File history File links Divisor_cube. ...
For other meanings of mathematics or math, see mathematics (disambiguation). ...
Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...
In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
The integers are commonly denoted by the above symbol. ...
In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...
In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ...
Modular form - Wikipedia /**/ @import /skins-1. ...
Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...
As an abstract term, congruence means similarity between objects. ...
In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ...
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The summatory function, with leading terms removed, for The summatory function, with leading terms removed, for The summatory function, with leading terms removed, for , graphed as a distribution or histogram. ...
Definition The divisor function σx(n) is defined as the sum of the xth powers of the positive divisors of n, or Addition is one of the basic operations of arithmetic. ...
In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
 The notations d(n) and τ(n) (the tau function) are also used to denote σ0(n), or the number of divisors of n. When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted.
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. | 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. |
Properties For a prime number p, In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ...
- d(p) = 2
- d(pn) = n + 1
- σ(p) = p + 1
because by definition, the factors of a prime number are 1 and itself. Clearly, 1 < d(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 In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...
In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...
 where r 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 In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...
Divisible is an Indie rock band from Los Angeles. ...
 which is equivalent to the useful formula:  An equation for calculating τ(n) is  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 τ(24) as so: | τ(24) |  | |  | The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, i.e. 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. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...
In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ...
In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ...
As an example, for two distinct primes p and q, let
- n = pq.
Then - φ(n) = (p − 1)(q − 1) = n + 1 − (p + q),
- σ(n) = (p + 1)(q + 1) = n + 1 + (p + q).
In 1984, Roger Heath-Brown proved that 1984 (MCMLXXXIV) was a leap year starting on Sunday of the Gregorian calendar. ...
(David Rodney) Roger Heath-Brown is a British mathematician, working in the field of analytic number theory. ...
- d(n) = d(n + 1)
will occur infinitely often.
Series expansion The divisor function can be written as a finite trigonometric series  without a explicit reference to the divisors of n, see Teilersumme.
Series relations Two Dirichlet series involving the divisor function are: In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...
 and  A Lambert series involving the divisor function is: A Lambert series, named after Johann Heinrich Lambert, is a series taking the form It can be resummed by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function Since this last sum is a typical number-theortic sum...
 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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ...
In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...
Approximate growth rate The behaviour of the sigma function is irregular. The growth rate of the sigma function can be expressed by:  where γ is Euler's constant. This result is Gronwall's theorem, published in 1913. The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...
Year 1913 (MCMXIII) was a common year starting on Wednesday (link will display the full calendar). ...
In 1984 Guy Robin proved that 1984 (MCMLXXXIV) was a leap year starting on Sunday of the Gregorian calendar. ...
- σ(n) < eγnloglogn for n > 5,040
is true if and only if the Riemann hypothesis is true (this is Robin's theorem). The largest known value that violates the inequality is n=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false then there are an infinite number of values of n that violate the inequality. Five thousand (5000) is the natural number following 4999 and preceding 5001. ...
Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...
A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that Jeffrey Lagarias is a professor at University of Michigan. ...
For album titles with the same name, see 2002 (album). ...
 for every natural number n, where Hn is the nth harmonic number. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
The harmonic number with (red line) with its asymptotic limit (blue line). ...
See also The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...
In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...
References - Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
- Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, volume 1, 1996, MIT Press. ISBN 0-262-02405-5, see page 234 in section 8.8.
- Robin, G. "Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann." J. Math. Pures Appl. 63, 187-213, 1984. Original publication of Robin's theorem.
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