DivisorSigma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed k

For fixed n

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[σ, k])(n) is an analytical function of k. (Subscript[σ, k])(n) is defined in the whole complex k -plane and for n∈ℤ .

Symmetries and periodicities

Parity

(Subscript[σ, k])(n) is an even function with respect to n.

Periodicity

No periodicity

Series representations

Other series representations

Jose Sousa (2007)

Product representations

Transformations

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Identities

Functional identities

Summation

Finite summation

Infinite summation

Asymptotic finite summation

Asymptotic infinite summation

Operations

Limit operation

Representations through equivalent functions

With related functions

Inequalities

Aardvark

Other identities

Congruence properties

Theorems

Inversion of one sum transformation

f(n)==(∑_(d|n))g((n/d))(Subscript[σ, 0])(d)⟺g(n)==(∑_(d|n))f((n/d))δ(d)/;   (δ(d)==(∏_(d|n))(({{2}, {(Subscript[n, k])}}))(((-1))^(Subscript[n, k]))/;d==(∏_(k=1)^r)((Subscript[p, k])^(Subscript[n, k])))

History

– R. Decartes (1638)
– J. Wallis (1658)
– E. Waring (1770)
– S. Ramanujan (1911) introduced the sum of the integer powers of divisors
– L. E. Dickson (1920) introduced symbol (Subscript[σ, k])(n)

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