Arithmetic function

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

number-theoretic 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 $\mathfrak U$.

Arithmetic functions appear and are employed in studies on the properties of numbers. However, the theory of arithmetic functions is also of independent interest. The laws governing the variations of arithmetic functions cannot usually be described by simple formulas, and the asymptotic behaviour in terms of numerical functions is determined. Since many arithmetic functions are not monotone, the study of their average values is of great importance (cf. Average order of an arithmetic function). An important class of arithmetic functions is constituted by the multiplicative arithmetic functions and the additive arithmetic functions. The problem of the distribution of their values is studied in probabilistic number theory [5].

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