Arithmetic functions

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An arithmetic function, also called an integer function or a number theoretic function, is a function $\scriptstyle a(n) \,$ defined for all positive integers $\scriptstyle n \,\in\, \mathbb{N}^+ \,$ , usually taken to be complex-valued, so that $\scriptstyle a:\, \mathbb{N} \,\to\, \mathbb{C} \,$ (Jones and Jones 1998, p. 143) and which typically expresses some arithmetical property of $\scriptstyle n \,$ .

Alternative definition of arithmetic function

An alternative definition of arithmetic function is a function $\scriptstyle \psi(n) \,$ s.t.

$\psi(m+n) = \psi(\psi(m) + \psi(n)) \,$

and

$\psi(mn) = \psi(\psi(m) \psi(n)) \,$

(Atanassov 1985; Trott 2004, p. 28).

Multiplicative and additive functions

An arithmetic function $\scriptstyle a(n) \,$ is

completely additive if $\scriptstyle a(m+n) \,=\, a(m) + a(n) \,$ for all positive integers $\scriptstyle m \,$ and $\scriptstyle n \,$ ;

completely multiplicative if $\scriptstyle a(mn) \,=\, a(m)a(n) \,$ for all positive integers $\scriptstyle m \,$ and $\scriptstyle n \,$ ;

Two positive integers $\scriptstyle m \,$ and $\scriptstyle n\,$ 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 $\scriptstyle a(n) \,$ is

additive if $\scriptstyle a(m+n) \,=\, a(m) + a(n) \,$ for all coprime positive integers $\scriptstyle m \,$ and $\scriptstyle n\,$ ;

multiplicative if $\scriptstyle a(mn) \,=\, a(m)a(n) \,$ for all coprime positive integers $\scriptstyle m \,$ and $\scriptstyle n\,$ .

Summatory functions

Given an arithmetic function $\scriptstyle a(n) \,$ , its summatory function $\scriptstyle A(x) \,$ is defined by

$A(x) := \sum_{n \le x} a(n). \,$

$\scriptstyle A(x) \,$ can be regarded as a function of a real variable $\scriptstyle x \,$ . Given a positive integer $\scriptstyle m \,$ , $\scriptstyle A(x) \,$ is constant along open intervals $\scriptstyle m \,<\, x \,<\, m + 1 \,$ , and has a jump discontinuity at each integer for which $\scriptstyle a(m) \,\neq\, 0 \,$ .

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

$A_0(m) := \frac12\left(\sum_{n < m} a(n) +\sum_{n \le m} a(n)\right) = A(m) - \frac12 a(m). \,$

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 $\scriptstyle x \,$ .

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: Springer-Verlag, pp. 143-162, 1998.

Arithmetic functions

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Contents

Alternative definition of arithmetic function

Multiplicative and additive functions

Summatory functions

References

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