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An arithmetic function, also called an integer function or a number theoretic function, is a function defined for all positive integers , usually taken to be complex-valued, so that (Jones and Jones 1998, p. 143) and which typically expresses some arithmetical property of .
Alternative definition of arithmetic function
An alternative definition of arithmetic function is a function s.t.
(Atanassov 1985; Trott 2004, p. 28).
Multiplicative and additive functions
An arithmetic function is
- completely additive if for all positive integers and ;
- completely multiplicative if for all positive integers and ;
Then an arithmetic function is
- additive if for all coprime positive integers and ;
- multiplicative if for all coprime positive integers and .
Given an arithmetic function , its summatory function is defined by
Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right
Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summatory functions "smooth out" these fluctuations. In some cases it may be possible to find the asymptotic behaviour for the summatory function for large .
- Atanassov, K., An Arithmetic Function and Some of Its Applications., Bull. Number Th. Related Topics 9, 18-27, 1985.
- Jones, G. A. and Jones, J. M., Arithmetic Functions., Ch. 8 in Elementary Number Theory. Berlin: Springer-Verlag, pp. 143-162, 1998.