Definition:

The Euler sigma function is defined as:
where the summation is over all positive divisors of .

Examples:

To illustrate how the Euler sigma function works, let’s compute for a few integers:

1. For ( n = 1 ):

• Divisors: ( 1 )
• 

2. For ( n = 6 ):

• Divisors: ( 1, 2, 3, 6 )
• 

3. For ( n = 12 ):

• Divisors: ( 1, 2, 3, 4, 6, 12 )
• 

4. For ( n = 28 ) (which is a perfect number):

• Divisors: ( 1, 2, 4, 7, 14, 28 )
• 

Properties:

1. Multiplicative: If and are coprime , then:
   
2. Relation to Prime Factorization: If the prime factorization of is given by:
   then the sigma function can be computed as: where: Thus,
3. Evenness: For , is always even, except when is a prime power, in which case can be odd.
4. Perfect Numbers: A positive integer is called a perfect number if . For example, the smallest perfect number is since .

Applications:

1. Number Theory: The sigma function plays a crucial role in divisor functions and their properties, such as in the study of perfect numbers and amicable numbers.
2. Analytic Number Theory: It is useful in understanding the distribution of divisors and has implications in results like Dirichlet series and zeta functions.
3. Combinatorial Mathematics: It is involved in counting problems and combinatorial identities.

Conclusion:

The Euler sigma function is a vital tool in number theory that provides insights into the structure and properties of integers through their divisors. Its multiplicative nature and relation to prime factorization make it particularly useful in various mathematical domains.