This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# DivisorSigma

 DivisorSigmagives the divisor function .
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• is the sum of the powers of the divisors of n.
Sum of divisors:
Sum of squares of divisors:
Plot the number of divisors:
Sums of divisors:
Sums of squared divisors:
 Out[1]=
Sum of divisors:
 Out[2]=
Sum of squares of divisors:
 Out[3]=

Plot the number of divisors:
 Out[1]=
Sums of divisors:
 Out[2]=
Sums of squared divisors:
 Out[3]=
 Scope   (2)
The first argument can be symbolic:
Include complex divisors:
 Options   (1)
By default, only integer divisors are included for integer input:
This also includes complex divisors:
 Applications   (4)
Plot the running average of the number of divisors with its asymptotic value:
Find :
Compute an iterated aliquot sum:
Compare the number of divisors with the totient:
Generate values using the definition:
Compute using DivisorSigma:
Use FullSimplify to simplify expressions containing DivisorSigma:
DivisorSigma is a multiplicative function:
Generating function:
With GaussianIntegers->True, the naive definition does not give the correct result:
To make DivisorSigma a multiplicative function, a definition involving factors is used:
Show the evolution of the limit :
Fourier transform of the divisor function in the complex plane:
New in 1