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Average Order

Values of many arithmetic functions fluctuate and so it is difficult to determine its behavior for large values of the argument.  

If typeset structure is an arithmetic function and typeset structure is a well behaved function (e.g. a simple elementary function) such that

lim _ (n -> ∞) (Underscript[∑, k <= n] f(k))/(Underscript[∑, k <= n] g(k)) = 1

then typeset structure is referred to as the average order of typeset structure.

Let

Overscript[f, ~](n) = 1/n Underoverscript[∑, k = 1, arg3] f(k) .

If

lim _ (n -> ∞) Overscript[f, ~](n)/g(n) = 1

then typeset structure is the average order of typeset structure.

Sometimes it is convenient to extend the range of considerations to the real numbers and to consider the limits of the form

lim _ (x -> ∞) (Underscript[∑, k <= x] f(k))/(Underscript[∑, k <= x] g(k)) .

The following generalization goes back to S.A. Amitsur [1], [2] :  Let typeset structure be the set of complex valued functions defined for typeset structure. Then given an arithmetic function typeset structure and typeset structure define the transform typeset structure by

(T _ f α) (x) = Underscript[∑, n <= x] f(n) α(x/n) .

If typeset structure for all typeset structure then typeset structure.

If we consider typeset structure as a vector space (with respect to the standard pointwise addition and scalar multiplications) then typeset structure is a linear map.

If typeset structure denotes the composition of functions in typeset structure and typeset structure the Dirichlet convolution of arithmetic functions, then

((T _ f o T _ g) α) (x) = Underscript[∑, n <= x] f(n) (Underscript[∑, k & ... 1;, m <= x] (Underscript[∑, k | m] f(k) g(m/k)) α(x/m) = (T _ (f * g) α) (x),

that is typeset structure. Similarly, typeset structure and typeset structure.

The set typeset structure forms a typeset structure-algebra with respect to the addition, composition and scalar multiplication. This typeset structure-algebra is isomorphic to the typeset structure-algebra of arithmetic functions.[3]  

References

[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.

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