Arithmetic function: Difference between revisions

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{{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''.

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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.

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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 ==

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{{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 ==

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''ϑ''(''x'') and ''ψ''(''x''), the [[Chebyshev function]]s, are defined as sums of the natural logarithms of the primes not exceeding ''x''.

<math display="block">\vartheta(x)=\sum_{p\le x} \lnlog p,</math>

<math display="block"> \psi(x) = \sum_{p^k\le x} \lnlog p.</math>

The second Chebyshev function ''ψ''(''x'') is the summation function of the von Mangoldt function just below.

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'''[[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 logarithm of the prime ''p'':

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

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

0&\text{if } n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;\text{ is not a prime power}.

\end{cases}</math>