Chapter: Arithmetic Functions

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.

Examples: 1

  1. Definition: Arithmetic Function
  2. Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors
  3. Theorem: Number of Multiples of a Prime Number Less Than Factorial
  4. Proposition: Calculating the Number of Positive Divisors
  5. Definition: Multiplicative Functions
  6. Section: Some Properties of the Möbius Function
  7. Section: An Application of the Möbius Inversion Formula
  8. Section: Dirichlet Convolution

Proofs: 1
Sections: 2
Theorems: 3

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927