Arithmetic function

A number theory or arithmetic function is a function that assigns to each positive natural number a function value of the complex numbers. These functions are used in number theory to describe properties of natural numbers, especially their divisibility and investigate.

  • 2.1 Definition
  • 2.2 Properties of the convolution
  • 2.3 Algebraic structure
  • 2.4 Demarcation of the space of the complex number sequences

Special number-theoretic functions

Examples

Important arithmetic functions are

  • The identical function and their powers,
  • The Dirichlet characters.
  • The divider number function d (n), which indicates how many divisors has the number n,
  • The generalized divisor sum functions
  • Euler's φ function, which specifies the number of prime to n natural numbers that are not greater than n,
  • The Liouville function
  • Dedekind psi function
  • The prime function π ( n ) indicating the number of prime numbers that are not greater than N,
  • The Möbius μ - function (see the paragraph on folding below) and
  • The p- adic exponential valuation
  • The divisor sum
  • The Smarandache function
  • The Chebyshev function
  • The Mangoldt function ( neither additive nor multiplicative)

Multiplicative functions

A number-theoretic function is called multiplicative, if a and b are always true for prime numbers and does not disappear. They called complete multiplicative, also strict or strictly multiplicative, if this also applies to non -prime numbers. Each completely multiplicative function is thus multiplicative. A multiplicative function can be represented as

  • Of the functions listed above as examples of the identities of their powers as well as the Dirichlet characters the divider number function, the divisor sum function and Euler's φ - function are completely multiplicative, multiplicative. The prime function and the exponential valuation are not multiplicative.
  • The ( pointwise ) product of two ( completely ) multiplicative functions is again ( completely ) multiplicative.

Additive functions

A number-theoretic function is called additive when a and b are always true for prime numbers. They called complete additive, also strict or strictly additive, if this also applies to non -prime numbers. An example of an additive feature is the p-adic exponent review. For any multiplicative function which vanishes nowhere, can be constructed an additive function by logarithmic earnings. Precisely: if F ( full), and is always multiplicatively, then a (fully) additive function. - Occasionally, a ( complex ) logarithm of a nowhere -vanishing number-theoretic function ( without amount) is formed. However, it is advised because of the various branches of the complex logarithm caution.

Convolution

The folding of number-theoretic functions is denoted by Dirichlet as Dirichlet convolution. For other meanings of the word in mathematics see the article convolution (mathematics).

Definition

The Dirichlet convolution of two number-theoretic functions defined by

Where the sum over all ( real and unreal, positive ) divisor of covers.

Summatorische the function of a number theoretic function is defined by, the constant function with the function value of 1 indicates, that

It can be shown that with regard to the folding operation can be inverted; its inverse is the ( multiplicative ) Möbius function μ. This leads to the Möbius inversion formula, with which you can recover a number-theoretic function from its summatorischen function.

Properties of the convolution

  • The convolution of two functions is the multiplicative multiplicative.
  • The convolution of two completely multiplicative functions need not be completely multiplicative.
  • Any number theoretic function f does not vanish at the point 1, has an inverse relation to the folding operation.
  • This convolution inverse is multiplicative if and only if f is multiplicative.
  • Folding completely multiplicative inverse of a multiplicative function is not generally completely multiplicative.
  • The neutral element of the convolution operation is the function η with η (1) = 1, η (n) = 0 for n> 1

Algebraic structure

  • The amount of number-theoretic functions together with the component-wise addition, scalar multiplication and convolution as multiplication of internal
  • A complex vector space,
  • An integral domain,
  • A commutative C- algebra.
  • The multiplicative group of this ring is made of the number-theoretic functions that do not vanish at the point 1.
  • The set of multiplicative functions is a proper subgroup of this group.

Distinguish from the space of the complex number sequences

With a complex scalar, componentwise addition and - instead of folding - the component-wise multiplication, the set of number-theoretic functions also forms a commutative C- algebra, the algebra of formal (not necessarily convergent ) complex number sequences. However, this canonical structure as picture space is in number theory of little interest.

As a complex vector space (ie without inner multiplication) is this sequence space with the space of number-theoretic functions identical.

Associated with Dirichlet series

Each number-theoretic function can be assigned to a formal Dirichlet series. The folding is then for multiplying rows. This construction is described in the article about Dirichlet series closer.

Prime number theorem Von Mangoldt function
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