Divisor function
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. Definition. The sum of positive divisors function σ("n"), for a real or complex number "z", is defined as the sum of the "z"th powers of the positive divisors of "n". It can be expressed in sigma notation as where formula_2 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 σ("n"), or the number-of-divisors function (). 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 σ("n") (). The aliquot sum "s"("n") of "n" is the sum of the proper divisors (that is, the divisors excluding "n" itself, ), and equals σ("n") − "n"; the aliquot sequence of "n" is formed by repeatedly applying the aliquot sum function. Example. For example, σ(12) is the number of the divisors of 12: while σ(12) is the sum of all the divisors: and the aliquot sum s(12) of proper divisors is: Table of values. The cases "x" = 2 to 5 are listed in − , "x" = 6 to 24 are  listed in − . 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 "p"# denotes the primorial, since "n" prime factors allow a sequence of binary selection (formula_8 or 1) from "n" terms for each proper divisor formed. Clearly, formula_9 for all formula_10, and formula_11 for all formula_12, formula_13 . The divisor function is multiplicative (since each divisor "c" of the product "mn" with formula_14 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 number of distinct prime factors of "n", "p" is the "i"th prime factor, and "a" is the maximum power of "p" by which "n" is divisible, then we have: which, when "x" ≠ 0, is equivalent to the useful formula: When "x" = 0, "d"("n") is: This result can be directly deduced from the fact that all divisors of formula_20 are uniquely determined by the distinct tuples formula_21 of integers with formula_22 (i.e. formula_23 independent choices for each formula_24). For example, if "n" is 24, there are two  prime factors ("p" is 2; "p" is 3); noting that 24 is the product of 2×3, "a" is 3 and "a" is 1. Thus we can calculate formula_25 as so: 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: where formula_28 if it occurs and formula_29 for formula_30, and formula_31 are consecutive pairs of generalized pentagonal numbers (, 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 formula_32 is even; for a square integer, one divisor (namely formula_33) is not paired with a distinct divisor and formula_32 is odd. Similarly, the number formula_35 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 is a power of 2, formula_36, then formula_37 and formula_38, which makes "n" almost-perfect. As an example, for two primes formula_39, let Then and where formula_45 is Euler's totient function. Then, the roots of express "p" and "q" in terms of "σ"("n") and "φ"("n") only, requiring no knowledge of "n" or formula_47, as Also, knowing and either formula_50 or formula_45, or, alternatively, formula_47 and either formula_50 or formula_45 allows an easy recovery of "p" and "q". In 1984, Roger  Heath-Brown proved that the equality is true for infinitely many values of , see . Series relations. Two Dirichlet series involving the divisor function are: which for "d"("n") = "σ"("n") gives: and a Ramanujan identity which is a special case of the Rankin–Selberg convolution. A Lambert series involving the divisor function is: 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 formula_60, there is an explicit series representation with Ramanujan sums formula_61 as : The computation of the first terms of  formula_63 shows its oscillations around the "average value" formula_64: Growth rate. In little-o notation, the divisor function satisfies the inequality: More precisely, Severin Wigert showed that: On the other hand, since there are infinitely many prime numbers, In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality: where formula_70 is Euler's gamma constant. Improving the bound formula_71 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 : where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 . His proof uses Mertens' 3rd theorem, which says that: where "p" denotes a prime. In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality: holds for all sufficiently large "n" . 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 . 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 . 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 . 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 formula_77 is the "n"th harmonic number, .