Arithmetic function: Difference between revisions

Revision as of 01:24, 16 January 2025

Latest revision as of 01:12, 6 April 2025

(6 intermediate revisions by 5 users not shown)

Line 1:

{{list|date=July 2020}}

{{short description|Function whose ___domain is the positive integers}}

{{log(x)}}

In [[number theory]], an '''arithmetic''', '''arithmetical''', or '''number-theoretic function'''<ref>{{harvtxt|Long|1972|p=151}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=58}}</ref> is generally any [[Function (mathematics)|function]] ''f''(''n'') whose [[Domain of a function|___domain]] is the set of [[natural number|positive integers]] and whose range is a [[subset]] of the [[complex number]]s.<ref>Niven & Zuckerman, 4.2.</ref><ref>Nagell, I.9.</ref><ref>Bateman & Diamond, 2.1.</ref> Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of ''n''".<ref>Hardy & Wright, intro. to Ch. XVI</ref> There is a larger class of number-theoretic functions that do not fit this definition, for example, the [[prime-counting function]]s. This article provides links to functions of both classes.

An example of an arithmetic function is the [[divisor function]] whose value at a positive integer ''n'' is equal to the number of divisors of ''n''.

Line 10 ⟶ 11:

An arithmetic function ''a'' is

* '''[[Completely additive function|completely additive]]''' if ''a''(''mn'') = ''a''(''m'') + ''a''(''n'') for all natural numbers ''m'' and ''n'';

* '''[[Completely multiplicative function|completely multiplicative]]''' if ''a''(1) = 1 and ''a''(''mn'') = ''a''(''m'')''a''(''n'') for all natural numbers ''m'' and ''n'';

Two whole numbers ''m'' and ''n'' are called [[coprime]] if their [[greatest common divisor]] is 1, that is, if there is no [[prime number]] that divides both of them.

Line 16 ⟶ 17:

Then an arithmetic function ''a'' is

* '''[[Additive function|additive]]''' if ''a''(''mn'') = ''a''(''m'') + ''a''(''n'') for all coprime natural numbers ''m'' and ''n'';

* '''[[Multiplicative function|multiplicative]]''' if ''a''(1) = 1 and ''a''(''mn'') = ''a''(''m'')''a''(''n'') for all coprime natural numbers ''m'' and ''n''.

== Notation ==

Line 44 ⟶ 45:

{{block indent | em = 1.5 | text = Ω(''n'') = ''a''₁ + ''a''₂ + ... + ''a''_''k''.}}

To avoid repetition, whenever possible formulas for the functions listed in this article are, whenever possible, given in terms of ''n'' and the corresponding ''p''_''i'', ''a''_''i'', ''ω'', and Ω.

== Multiplicative functions ==

Line 155 ⟶ 156:

=== Λ(''n'') – von Mangoldt function ===

'''[[von Mangoldt function|Λ(''n'')]]''', the von Mangoldt function, is 0 unless the argument ''n'' is a prime power {{math|''p''^''k''}}, in which case it is the natural loglogarithm of the prime ''p'':

<math display="block">\Lambda(n) = \begin{cases}

\log p &\text{if } n = 2,3,4,5,7,8,9,11,13,16,\ldots=p^k \text{ is a prime power}\\