Arithmetic function: Difference between revisions

Revision as of 23:39, 26 November 2022

Revision as of 23:40, 26 November 2022

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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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| isbn = 978-0-8218-2076-6}}

* {{citation | last=Williams | first=Kenneth S. | title=Number theory in the spirit of Liouville | zbl=1227.11002 | series=London Mathematical Society Student Texts | volume=76 | ___location=Cambridge | publisher=[[Cambridge University Press]] | isbn=978-0-521-17562-3 | year=2011 }}

* {{cite encyclopedia

|title= Integer Function

|encyclopedia= MathWorld

|last= Weisstein

|first= Eric S.

|publisher= Wolfram Research

|url= https://mathworld.wolfram.com/IntegerFunction.html

|date= 2003-07-14

|access-date= 2022-11-26

}}

==Further reading==