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Sigma Functions in Number Theory 📂Number Theory

Sigma Functions in Number Theory

Theorem

For σ(n):=dnd\displaystyle \sigma (n) : = \sum_{d \mid n} d, the following holds:

  • [1]: For a prime number pp, σ(pk)=pk+11p1\sigma ( p^k ) = {{p^{k+1} - 1} \over {p-1}}
  • [2]: If gcd(n,m)=1\gcd (n , m ) = 1, then σ(nm)=σ(n)σ(m)\sigma (nm) = \sigma (n) \sigma (m)

Description

The sigma function, simply put, is the sum of divisors, for example, for 66, it is σ(6)=1+2+3+6=12\sigma (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 nn that satisfies σ(n)=2n\sigma (n) = 2n can be defined as a perfect number.

Proof

[1]

σ(pk)=1+p++pk=pk+11p1 \sigma ( p^k ) = 1 + p + \cdots + p^{k} = {{p^{k+1} - 1} \over {p-1}}

[2]

Let’s denote the divisors of nn as 1,dn1,dn2,,dnN,n1, d_{n1}, d_{n2}, \cdots, d_{nN}, n, and the divisors of mm as 1,dm1,dm2,,dmM,m1, d_{m1}, d_{m2}, \cdots, d_{mM}, m.

Since gcd(n,m)=1\gcd(n,m) = 1, dnmd=1+dn1+dm1+dn1dm1++nm \sum_{d \mid nm} d = 1 + d_{n1} + d_{m1} + d_{n1} d_{m1} + \cdots + nm , to sum up, dnmd=(1+dn1++n)(1+dm1++m)=dndndmdm \sum_{d \mid nm} d = (1 + d_{n1} + \cdots + n ) (1 + d_{m1} + \cdots + m) = \sum_{d | n} d_{n} \sum_{d | m} d_{m}