Arithmetic function: Difference between revisions

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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 is the [[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 exampleExamples of an arithmetic functionfunctions isinclude the [[divisor function]] whose value at a positive integer ''n'' is equal to the number of divisors of ''n''., the '''unit function''' ''u(n)'', defined by ''u(n) = 1'' for all ''n ∈ ℤ⁺'', and the '''identity function''' ''I(n)'', defined as:

<math>

I(n) =

\begin{cases}

1, & \text{if } n = 1, \\

0, & \text{otherwise.}

\end{cases}

</math>

Arithmetic functions are often extremely irregular (see [[#First 100 values of some arithmetic functions|table]]), but some of them have series expansions in terms of [[Ramanujan's sum]].