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{{short description|Function whose ___domain is the positive integers}}
In [[number theory]], an '''arithmetic''', '''arithmetical''', '''integer''' or '''number-theoretic function'''<ref>{{harvtxt|Long|1972|p=151}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=58}}</ref><ref>{{harvtxt|Weisstein|2003|}}</ref> is for most authors<ref>Niven & Zuckerman, 4.2.</ref><ref>Nagell, I.9.</ref><ref>Bateman & Diamond, 2.1.</ref> 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. 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>
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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