Example: Examples of Important Arithmetic Functions

(related to Chapter: Arithmetic Functions)

In this sense, $f(n)=n!$, $f(a)=\cos(n)$, $f(n)=\frac 1n$, $f(n)=\exp(in)$ are all arithmetic functions, but only some arithmetic functions are of particular interest for number theorists. We will now introduce some of these important arithmetic functions. The reader is invited to play around with the provided interactive examples. At this stage, it is only important to mention that the graphs of most of the introduced number-theoretic functions look very "weird" and it is hard to spot any particular patterns. In order to get the reader more acquainted with these functions, we will only introduce them, without going deeper into their particularly interesting number theoretic properties. This will happen later in the text.

  1. Definition: Prime-Counting Function
  2. Definition: Number of Divisors
  3. Definition: Möbius Function, Square-free
  4. Definition: Euler function
  5. Definition: Von Mangoldt Function

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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