Arithmetic function: Difference between revisions

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Formally, in [[number theory]], an '''arithmetic function''' is simply a sequence, with real or complex values. A sequence is, of course, a [[function]] on the set of [[natural number]]s (i.e. [[positive]] [[integer]]s). To emphasize that we are thinking of them as functions, we shall usually use notation like ''a(n)'', rather than ''a_n'', for the value corresponding to the integer ''n''. The term arithmetic function is used especially when ''a(n)'' is defined using number-theoretic properties in some way. A large part of number theory consists, in one way or another, of the study of these functions. <ref>

In [[number theory]] and [[computability theory]], subfields of [[mathematics]], a '''number-theoretic function''' is any [[function (mathematics)|function]] whose [[___domain (mathematics)|___domain]] is the set of [[natural number]]s.<ref>{{cite book

|title=Fundamentals of Number Theory

|author=William J. LeVeque

|year=1996

|publisher=Courier Dover Publications

|isbn=0486689069}}<br/>

{{cite book

|title=IntroductionThe toPrime MathematicalNumber Theorem Logic

|author=ElliottG. J. O. Jameson Mendelson

|year=19872003

|publisher=CRCLondon PressMathematical Society

|isbn=04128083070-521-89110-8}}</ref>

A number-theoretic function whose range is included in the set of [[complex number]]s is called an '''arithmetical function''' or '''arithmetic function'''.<ref>

{{cite book

|title=Elementary Number Theory: A Computer Approach

|author= Allan M. Kirch

|year=1974

|publisher=Intext Educational Publishers

|isbn=0700224564}}<br/>

{{cite book

|title=Classical Theory of Arithmetic Functions

|author=R. Sivaramakrishnan and Sivaramakrishnan Sivaramakrishnan

|year=1988

|publisher=Marcel Dekker

|isbn= 0824780817}}</ref>

The most important arithmetic functions are the [[additive function|additive]] and the [[multiplicative function|multiplicative]] ones. An important operation on arithmetic functions is the [[Dirichlet convolution]]. Arithmetic functions may be studied with [[Bell series]].

Formally, an '''arithmetic function''' is simply a sequence, with real or complex values. A sequence is, of course, a [[function]] on the set of [[natural number]]s (i.e. [[positive]] [[integer]]s). To emphasize that we are thinking of them as functions, we shall usually use notation like ''a(n)'', rather than ''a_n'', for the value corresponding to the integer ''n''. The term arithmetic function is used especially when ''a(n)'' is defined using number-theoretic properties in some way. A large part of number theory consists, in one way or another, of the study of these functions.

==Examples==