A fundamental concept of number theory is the concept of arithmetic functions, also known as number-theoretic functions. Arithmetic functions give a toolset for the study of the prime numbers, which on one hand are a “simple” series of natural numbers, on the other hand, however, a series which is extremely hard to be described. For instance, no simple function (formula) is known, which would “calculate”, whether or not a given number $n$ is a prime number or not.
Number theorists try to develop tools which are, from the very beginning, capable to deal with such complicated phenomena like prime numbers. Therefore, they decided to define arithmetic functions not as mappings $f:\mathbb N \to\mathbb N$ of natural numbers to natural numbers, but be more general, as mappings of naturals numbers to complex numbers. In other words, in the general case, arithmetic functions can be identified with a series of complex numbers. In most cases, however, the series of function values created by a given arithmetic function will consist of pure real, or even integer or natural numbers.
| | | | created: 2014-02-21 00:07:13 | modified: 2019-03-21 07:34:18 | by: bookofproofs | references: , 
 Landau, Edmund: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927
 Scheid Harald: “Zahlentheorie”, Spektrum Akademischer Verlag, 2003, 3. Auflage