Multiplicative Function
A multiplicative function is an arithmetic function satisfying for coprime integers and A completely multiplicative function satisfies for all values of and
Multiplicative functions arise naturally in many contexts in number theory and algebra. The Dirichlet series associated with multiplicative functions have useful product formulas, such as the formula for the Riemann zeta function.
Contents
Important Multiplicative Functions
Here are important multiplicative functions:
- Euler's totient function: the number of positive integers such that gcd
- The Möbius function:
- The sum of powers of divisors:
- The function
- The unit function:
- The identity function:
These last three are completely multiplicative as well.
Values on Prime Powers
If is a multiplicative function, then it is completely determined by its values on prime powers. If is completely multiplicative, then it is completely determined by its values on primes.
This follows from unique factorization: write for some nonnegative , primes , and positive integers . Then
because the prime powers are all relatively prime to each other. So is the product of the values of . If is completely multiplicative, then , so is determined by the values of .
Connection with Dirichlet Convolution
If is a multiplicative function, then it is completely determined by its values on prime powers. If is completely multiplicative, then it is completely determined by its values on primes.
This follows from unique factorization: write for some nonnegative , primes , and positive integers . Then
because the prime powers are all relatively prime to each other. So is the product of the values of . If is completely multiplicative, then , so is determined by the values of .
Examples and Applications
The techniques of the first theorem can be used to write product formulas for multiplicative functions. For instance, consider the multiplicative function When this evaluates to , so
Letting , the following formulas can be derived in a similar way:
The two theorems taken together can be used to derive several interesting identities involving multiplicative functions. The first theorem implies that two multiplicative functions that agree on prime power arguments must be the same function. This is useful for proofs.
Let the number of distinct prime factors of . Let the product of the distinct prime factors of . Show that
Proof: The product of multiplicative functions is clearly multiplicative, so is multiplicative. The sum on the left is the Dirichlet convolution , so it is multiplicative by the second theorem. The product on the right is also multiplicative, as it is easy to check that both and are multiplicative functions.
So it suffices to show the theorem is true for , prime. In this case the left side is and the right side is , so the two sides agree on prime powers and hence agree everywhere.
Show that
Proof: Both sides are multiplicative (similarly to the previous example), so it is enough to check that they agree for . In this case, the left side is
and the right side is
but it is easy to check that these are equal.