Arithmetic function: Difference between revisions

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In [[number theory]], an '''arithmetic function''' (or '''number-theoretic function''') ''f''(''n'') is a [[function]] defined for all positive [[integer|integers]] and having values in the [[complex number|complex numbers]] (''f'' : '''Z'''⁺ &equiv; '''N''' <font size=+1 color=darkgreen> &rarr; </font> '''C''').

Another definition of arithmetic function is a function with exactly two [[composition|compositional properties]]:

The most important arithmetic functions are the [[multiplicative function|multiplicative]] ones.

:&Psi;(''m''+''n'') = &Psi;[&Psi;(''m'')+&Psi;(''n'')] <tt> /composition with addition/ </tt>

:&Psi;(''m'',''n'') = &Psi;[&Psi;(''m'')&Psi;(''n'')] <tt> /composition with parameter/ </tt>

The most important arithmetic functions are the [[additive function|additive]] and the [[multiplicative function|multiplicative]] ones.

'''Note:''' Arithmetic function should not be confused with a function sometimes called ''arithmetical function'' which is in fact [[integer function]] (''f'' : '''N''' <font size=+1 color=darkgreen> &rarr; </font> '''N''').