📂Number Theory

Arithmetic Functions in Analytic Number Theory

Definition ¹

A function whose domain is the set of natural numbers $\mathbb{N}$ and whose range is the set of real numbers $\mathbb{R}$ or the set of complex numbers $\mathbb{C}$ is called an arithmetic function.

Description

In analytic number theory, there is interest in the properties and relationships of various arithmetic functions, including examples such as:

1. Identity function $I$
2. Divisor function $\sigma_{\alpha}$
3. Norm $N$
4. Divisor function $\sigma_{\alpha}$
5. Möbius function $\mu$
6. Euler’s totient function $\varphi$
7. Unit function $u$
8. Mangoldt function $\Lambda$
9. Liouville function $\lambda$

The definition of an arithmetic function might not seem new—it’s essentially just a sequence. Indeed, sequences have always been functions, though in many areas of mathematics, the term itself is commonly used distinct from functions. However, in (analytic) number theory, since the subjects of interest are naturally numbers, having $\mathbb{N}$ or $\mathbb{Z}$ as the domain is sufficient, and there’s hardly a reason to distinguish between a sequence and a function. However, since they are treated more closely as functions, the term arithmetic function is used. Formally, even though the domain is not a vector field, the fact that the range is $\mathbb{R}$ or $\mathbb{C}$ is similar to functionals.

Furthermore, there is interest in the series of arithmetic functions. For example, for a given arithmetic function $f$, finding $\displaystyle F(n) = \sum_{d \mid n} f(d)$. $\displaystyle \sum_{d \mid n}$ involves calculations for all divisors $d$ of $n$, which can be understood in a similar sense to calculating $\displaystyle \sum_{k=1}^{\infty}$ in general analysis.