Average order of an arithmetic function: Difference between revisions

Revision as of 23:03, 5 August 2015

Revision as of 16:48, 13 August 2015

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Thus, (''a'', ''b'') is visible from (0,0) if and only if gcd(''a'', ''b'') = 1.

Notice that <math>\frac{\varphi(n)}{n}</math> is the probability of a random point on the square <math>\{(r,s)\in \mathbb{N} : \max(|r|,|s|)=n\}</math> to be visible from the origin.

Thus, one can show that the natural density of the points which are visible from the origin is given by the average,

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<math>\zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\text{deg(f)=n}}q^{-sn}=\sum_{n}(q^{n-sn})=(1-q^{1-s})^{-1}. </math>

In a similar way, If ''&fnof;ƒ'' and ''g'' are two polynomial arithmetic functions, one defines ''ƒ''&nbsp;*&nbsp;''g'', the ''Dirichlet convolution'' of ''ƒ'' and ''g'', by

:<math>

\begin{align}

(f*g)(m)

&= \sum_{d\,\mid \,m} f(m)g\left(\frac{m}{d}\right) \\

&= \sum_{ab\,=\,m}f(a)g(b)

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: <math>D(n)=x \log_q(x)+x</math> which resembles closely the analogous result for integers <math>\sum_{k \mathop =1}^n d(k)=x\log x+(2\gamma-1) x + O(\sqrt{x})</math>, where <math>\gamma</math> is [[Euler constant]].

It is interesting to note that not a lot is known about the error term for the integers, while in the polynomials case, there is no error term!

This is because of the very simple nature of the zeta function <math>\zeta_{A}(s)</math>, and that it has NO zeros.

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and we get that,

: <math>\zeta_{A}(s)D_{\Lambda_{A}}(s)=\sum_{m}log|m||m|^{-s}.</math>

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|author=Tom M. Apostol

|year=1976

|publisher=Springer [[Undergraduate Texts in Mathematics]]

|isbn=0-387-90163-9}}

* {{Citation