Mobius function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
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Definition

The Mobius function is an integer-valued function defined on the natural numbers as follows. The Mobius function at $n$, denoted $\mu (n)$, is defined as:

• $\mu (1)=1$.
• $\mu (p_{1}p_{2}\dots p_{r})=(-1)^{r}$ if $p_{i}$ are pairwise distinct primes.
• $\mu (n)=0$ if $n$ is divisible by the square of a prime.

Definition in terms of Dirichlet product

The Mobius function $\mu$ is defined as the inverse, with respect to the Dirichlet product, of the all ones function $U$, which is defined as the function sending every natural number to $1$. In other words:

$\mu *U=U*\mu =I$.

Here, $I$ is the identity element for the Dirichlet product, and is the function that is $1$ at $1$ and $0$ elsewhere.

Dirichlet series

The Dirichlet series of the Mobius function is given by:

$\sum _{n\in mathbb{N}}{\frac {\mu (n)}{n^{s}}}$.

This is equal to the reciprocal of the Riemann zeta-function, because the Mobius function is the inverse of the all ones function with respect to the Dirichlet product. In particular, the function is absolutely convergent for $\operatorname {Re} (s)>1$, and it has an analytic continuation to $\mathbb {C}$, with its poles being the zeros of the Riemann zeta-function and its zeros being the poles of the Riemann zeta-function.

Facts

Mobius inversion formula

Further information: Mobius inversion formula

In terms of Dirichlet products, the Mobius inversion formula states that:

$f*U=g\iff f=g*\mu$.

The group-theoretic proof of this involves taking the Dirichlet product of both sides with $\mu$.

In more explicit terms, it states that:

$g(n)=\sum _{d|n}f(d)\ \forall \ n\in \mathbb {N} \iff f(n)=\sum _{d|n}g(d)\mu (n/d)\ \forall \ n\in \mathbb {N}$.