In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose domain is the set of natural numbers. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an ^[1]

A number-theoretic function whose range is included in the set of complex numbers is called an arithmetical function or arithmetic function. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted ^[2] The most important arithmetic functions are the additive and the multiplicative ones. Different definitions exist depending on the specific field of application Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative An important operation on arithmetic functions is the Dirichlet convolution. In Mathematics, the Dirichlet convolution is a Binary operation defined for Arithmetic functions it is of importance in Number theory. Arithmetic functions may be studied with Bell series. In Mathematics, the Bell series is a Formal power series used to study properties of arithmetical functions

Examples

The articles on additive and multiplicative functions contain several examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:

• r₄(n) - the number of ways that n can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

1 = 1²+0²+0²+0² = 0²+1²+0²+0² = 0²+0²+1²+0² = 0²+0²+0²+1²,

hence r₄(1)=4.

• P(n), the Partition function - the number of representations of n as a sum of positive integers, where we don't distinguish between different orders of the summands. In Number theory, a partition of a positive Integer n is a way of writing n as a Sum of positive integers For instance: P(2 · 5) = P(10) = 42 and P(2)P(5) = 2 · 7 = 14 ≠ 42.

• π (n), the Prime counting function - the number of primes less than or equal to a given number n. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 We have π(1) = 0 and π(10) = 4 (the primes below 10 being 2, 3, 5, and 7).

• ω (n), the number of distinct primes dividing given number n. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 We have ω(1) = 0 and ω(20) = 2 (the distinct primes dividing 20 being 2 and 5).

• Λ(n), the von Mangoldt function which is defined to be ln(p) if n is an integer power of a prime p and 0 for all other n. In Mathematics, the von Mangoldt function is an Arithmetic function named after German mathematician Hans von Mangoldt.

Footnotes

1. ^ William J. LeVeque (1996). Fundamentals of Number Theory. Courier Dover Publications. ISBN 0486689069.  
   Elliott Mendelson (1987). Introduction to Mathematical Logic. CRC Press. ISBN 0412808307.  
2. ^ Allan M. Kirch (1974). Elementary Number Theory: A Computer Approach. Intext Educational Publishers. ISBN 0700224564.  
   R. Sivaramakrishnan and Sivaramakrishnan Sivaramakrishnan (1988). Classical Theory of Arithmetic Functions. Marcel Dekker. ISBN 0824780817.  

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