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Values of many arithmetic functions fluctuate and so it is difficult to determine its behavior for large values of the argument.
If is an arithmetic function and
is a well behaved function (e.g. a simple elementary function) such that
then is referred to as the average order of
.
Let
If
then is the average order of
.
Sometimes it is convenient to extend the range of considerations to the real numbers and to consider the limits of the form
The following generalization goes back to S.A. Amitsur [1], [2] : Let be the set of complex valued functions defined for
. Then given an arithmetic function
and
define the transform
by
If for all
then
.
If we consider as a vector space (with respect to the standard pointwise addition and scalar multiplications) then
is a linear map.
If denotes the composition of functions in
and
the Dirichlet convolution of arithmetic functions, then
that is . Similarly,
and
.
The set forms a
-algebra with respect to the addition, composition and scalar multiplication. This
-algebra is isomorphic to the
-algebra of arithmetic functions.[3]
[1] | Amitsur, S. A. (1961). Arithmetic linear transformations and abstract prime number theorems. Canad. J. Math., 13, 83-109. |
[2] | Amitsur, A. S. (1969). Corrig.. Canad. J. Math., 21, 1-5. |
[3] | Scheid, H. (1994). Zahlentheorie (2nd revised ed.). Mannheim Leipzig Wien Zürich: BI Wissenschaftsverlag. |
Cite this web-page as:
Štefan Porubský: Average Order.