Arithmetic function in the context of "Number theory"

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Big O notation is a mathematical notation that describes the approximate size of a function on a domain. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann and Edmund Landau and expanded by others, collectively called Bachmann–Landau notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation.

In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express bounds on the growth of an arithmetical function; one well-known example is the remainder term in the prime number theorem.In mathematical analysis, including calculus,Big O notation is used to bound the error when truncating a power series and to express the qualityof approximation of a real or complex valued functionby a simpler function.

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function ${\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} }$ is a Dirichlet character of modulus ${\displaystyle m}$ (where ${\displaystyle m}$ is a positive integer) if for all integers ${\displaystyle a}$ and ${\displaystyle b}$:

The simplest possible character, called the principal character and usually denoted ${\displaystyle \chi _{0}}$, exists for all moduli: