Sigma Functions in Number Theory
📂Number TheorySigma Functions in Number Theory
Theorem
For σ(n):=d∣n∑d, the following holds:
- [1]: For a prime number p,
σ(pk)=p−1pk+1−1
- [2]: If gcd(n,m)=1, then
σ(nm)=σ(n)σ(m)
Description
The sigma function, simply put, is the sum of divisors, for example, for 6, it is σ(6)=1+2+3+6=12. In analytic number theory, it is generalized as a divisor function.
Additionally, by mentioning the sigma function, it allows for a clean definition of Perfect Number. A perfect number is a number where the sum of its proper divisors (excluding itself) is equal to the number itself. Thus, a number n that satisfies σ(n)=2n can be defined as a perfect number.
Proof
[1]
σ(pk)=1+p+⋯+pk=p−1pk+1−1
■
[2]
Let’s denote the divisors of n as 1,dn1,dn2,⋯,dnN,n, and the divisors of m as 1,dm1,dm2,⋯,dmM,m.
Since gcd(n,m)=1,
d∣nm∑d=1+dn1+dm1+dn1dm1+⋯+nm,
to sum up,
d∣nm∑d=(1+dn1+⋯+n)(1+dm1+⋯+m)=d∣n∑dnd∣m∑dm
■