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An
arithmetic function, also called an integer function or a numbertheoretic function, is a function
defined for all
positive integers , usually taken to be
complexvalued (Jones and Jones 1998, p. 143).
Some authors (Atanassov 1985; Trott 2004, p. 28) use this term in a nonstandard sense to describe functions
s.t.

ψ (m + n) = ψ (ψ (m) + ψ (n)) 
and

ψ (m n) = ψ (ψ (m) ψ (n)). 
The remainder of this page does not use this definition.
Multiplicative and additive functions
An arithmetic function
is
Two positive integers
and
are called
coprime if their
greatest common divisor is
1; i.e., if there is no
prime number that divides both of them.
Then an arithmetic function
is
 additive if
a (m + n) = a (m) + a (n) 
for all coprime positive integers and ;
Summatory functions
Given an arithmetic function
, its
summatory function is defined by

can be regarded as a function of a real variable
. Given a positive integer
,
is constant along open intervals
, and has a
jump discontinuity at each integer for which
.
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

A0(m) := a (n) + a (n) = A (m) − a (m). 
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
.
References
 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: SpringerVerlag, pp. 143–162, 1998.