Arithmetic functions and Dirichlet convolution

An arithmetic function is any map from the positive integers to the complex numbers. Define its Dirichlet convolution with another function by summing over positive divisors. The divisor paired with a chosen divisor is the complementary factor, so exchanging the two factors proves commutativity. Summing over ordered triples of factors proves associativity. Every sum here is finite; neither convergence estimates nor an ordering of infinitely many terms is needed for these algebraic identities.

Write $\mathbf{1}(n)=1$ for the constant function, and $\varepsilon(1)=1$, $\varepsilon(n)=0$ for larger integers. The convolution identity is epsilon, not the constant function: only the divisor with complementary factor one survives. A function has a convolution inverse precisely when its value at one is nonzero. Solve first for the inverse value at one, then recursively isolate the inverse value at each larger integer. This construction also proves uniqueness.

A multiplicative function satisfies $f(1)=1$ and $f(mn)=f(m)f(n)$ when the two arguments are coprime. Completely multiplicative functions satisfy the latter identity without that restriction. These are different conditions: the divisor count is multiplicative, but its values at a prime and its square are two and three, rather than two and four. Unique factorization reduces a multiplicative function to its values at prime powers, where repeated occurrences of the same prime must still be handled explicitly.

The Möbius function is zero on integers divisible by a prime square and equals $(-1)^r$ on a product of distinct primes. Its divisor sum is one at the integer one and zero elsewhere: for an integer with distinct prime factors, the sum expands the product of one minus one over those factors. Consequently it is the convolution inverse of the constant function. If a divisor sum obscures an unknown arithmetic function, convolving with Möbius recovers that function without approximation.

Let $\operatorname{id}(n)=n$. Sorting the integers from one through a fixed integer by their greatest common divisor with it gives $\sum_{d\mid n}\varphi(d)=n$. Möbius inversion therefore expresses the totient as $\mu*\operatorname{id}$. Factoring this finite sum prime by prime gives the familiar product of the integer with one minus the reciprocal of each distinct prime factor. The formula counts units, while convolution explains why its factors appear and why prime powers require only one factor each.

Another useful function is $\Lambda(n)$, equal to the natural logarithm of a prime when the integer is a positive power of that prime, and zero otherwise, including at one. Summing these weights over divisors counts each occurrence of each prime in the factorization, so $\mathbf{1}*\Lambda=\log$. Thus $\Lambda=\mu*\log$. This isolates prime-power information from a smooth-looking logarithm and prepares the logarithmic derivative of an Euler product. It does not assert that the weight is supported only on primes.