# Arithmetic function

A complex-valued function, the domain of definition of which is one of the following sets: the set of natural numbers, the set of rational integers, the set of integral ideals of a given algebraic number field, a lattice in a multi-dimensional coordinate space, etc. These are arithmetic functions in the wide sense. However, the term is often employed to denote a function of the above type with special arithmetic properties. The most commonly occurring arithmetic functions have traditional symbolic notations: $\phi(n)$ is the Euler function; $d(n)$ or $\tau(n)$ is the number of divisors; $\mu(n)$ is the Möbius function; $\Lambda(n)$ is the Mangoldt function; $\sigma(n)$ is the sum of divisors of the number $n$. Arithmetic functions also include the integral part of a number, $[x]$, and the fractional part of a number, $\{x\}$. Arithmetic functions giving the number of solutions of an equation are also studied; for example, $r(n)$ is the number of integer solutions $x$ and $y$ of the equation $x^2+y^2=n$ in the Goldbach problem; $J(N)$ is the number of solutions in prime numbers of the equation $N=p_1+p_2+p_3$. Other arithmetic functions express the quantity of numbers satisfying certain conditions; thus, for instance, the function $\pi(x)$ — the number of primes not larger than $x$ — describes the distribution of primes; $\pi(x,q,l)$ gives the number of primes not larger than $x$ in the arithmetic progression $p\equiv l\pmod q$. The Chebyshev functions also deal with properties of primes: $\theta(x)$ is the sum of the natural logarithms of the prime numbers up to $x$, while $\psi(x)=\sum_{n\leq x}\Lambda(n)$ (cf. Chebyshev function).
Algebraic number theory deals with generalizations of the above arithmetic functions of a natural argument. Thus, for instance, in an algebraic field $K$ of degree $n$, Euler's function $\phi(\mathfrak U)$ — the number of residue classes by the ideal $\mathfrak U$ mutually prime with $\mathfrak U$ — is introduced for an integral ideal $\mathfrak U$.