Average order of an arithmetic function: Difference between revisions

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Examples: The term was not matching the definition above (probably it was taken from Apostol’1976 §3.9 which uses different terms).
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In cases where the limit,
 
: <math>\lim_{n\rightarrowto\infty}\text{Ave}_n(h) = c</math>
 
exists, it is said that ''h'' has a '''mean value''' ('''average value''') ''c''.
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Let ''h'' be a polynomial arithmetic function (i.e. a function on set of monic polynomials over ''A''). Its corresponding Dirichlet series define to be
 
: <math>D_{h}(s)=\sum_{f\text{ monic}} h(f)|f|^{-s},</math>
 
where for <math>g\in A</math>, set <math>|g|=q^{deg(g)}</math> if <math>g\ne 0</math>, and <math>|g|=0</math> otherwise.
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The polynomial zeta function is then
 
: <math>\zeta_{A}(s)=\sum_{f\text{ monic}} |f|^{-s}.</math>
 
Similar to the situation in {{math|'''N'''}}, every Dirichlet series of a [[multiplicative function]] ''h'' has a product representation (Euler product):
 
: <math>D_{h}(s)=\prod_{P}(\sum_{n\mathop =0}^{\infty} h(P^{n})|P|^{-sn}),</math>
 
Where the product runs over all monic irreducible polynomials ''P''.
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Unlike the classical [[zeta function]], <math>\zeta_{A}(s)</math> is a simple rational function:
<math display="block">\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 ''ƒf'' and ''g'' are two polynomial arithmetic functions, one defines ''ƒf''&nbsp;*&nbsp;''g'', the ''Dirichlet convolution'' of ''ƒf'' and ''g'', by
<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>
 
:<math>\begin{align}
In a similar way, If ''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)
\end{align}</math>
</math>
 
where the sum extends over all monic [[divisor]]s ''d'' of&nbsp;''m'', or equivalently over all pairs (''a'', ''b'') of monic polynomials whose product is ''m''. The identity <math>D_h D_g =D_{h*g}</math> still holds. Thus, like in the elementary theory, the polynomial Dirichlet series and the zeta function has a connection with the notion of mean values in the context of polynomials. The following examples illustrate it.
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Finally, expand the left-hand side in a geometric series and compare the coefficients on <math>u^{n}</math> on both sides, to conclude that
 
<math display="block">b_{n}=\begin{cases}
\;\;\,q^{n} & n\le k-1 \\
\;\;\, q^{n}(1-q^{1-k}) &\text{otherwise}
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Hence,
<math display="block">\text{Ave}_{n}(\delta) = 1-q^{1-k} = \frac{1}{\zeta_{A}(k)}</math>
 
<math>\text{Ave}_{n}(\delta)=1-q^{1-k}=\frac{1}{\zeta_{A}(k)}</math>
 
And since it doesn't depend on ''n'' this is also the mean value of <math>\delta(f)</math>.
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First, notice that
<math display="block">\sigma_{k}(m)=h*\mathbb{I}(m)</math>
 
:where <math>\sigma_{k}h(mf)=h*|f|^{k}</math> and <math>\mathbb{I}(mf)=1\;\; \forall{f}</math>.
 
where <math>h(f)=|f|^{k}</math> and <math>\;\mathbb{I}(f)=1\;\; \forall{f}</math>.
 
Therefore,
 
: <math>\sum_{m}\sigma_{k}(m)|m|^{-s}=\zeta_{A}(s)\sum_{m}h(m)|m|^{-s}.</math>
 
Substitute <math>q^{-s}=u</math> we get,
 
: <math>\text{LHS}=\sum_n(\sum_{\deg(m)=n} \sigma_k(m))u^n</math>, and by [[Cauchy product]] we get,
 
: <math>\begin{align}
\begin{align}
\text{RHS}
&=\sum_n q^{n(1-s)}\sum_{n}(\sum_{\deg(m)=n}h(m))u^n \\
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&=\sum_n (\sum_{j \mathop =0}^{n}q^{n-j}q^{jk+j}) \\
&=\sum_n (q^n(\frac{1-q^{k(n+1)}}{1-q^k})) u^n.
\end{align}</math>
</math>
 
Finally we get that,
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Notice that
 
: <math>q^n \text{Ave}_n\sigma_{k} = q^{n(k+1)}(\frac{1-q^{-k(n+1)}}{1-q^{-k}}) = q^{n(k+1)}(\frac{\zeta(k+1)}{\zeta(kn+k+1)})</math>
 
Thus, if we set <math>x=q^n</math> then the above result reads
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which resembles the analogous result for the integers:
<math display="block">\sum_{n\le x}\sigma_{k}(n)=\frac{\zeta(k+1)}{k+1}x^{k+1}+O(x^{k})</math>
 
<math>\sum_{n\le x}\sigma_{k}(n)=\frac{\zeta(k+1)}{k+1}x^{k+1}+O(x^{k})</math>
 
====Number of divisors====
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Let <math>d(f)</math> be the number of monic divisors of ''f'' and let <math>D(n)</math> be the sum of <math>d(f)</math> over all monics of degree n.
 
<math display="block">\zeta_A(s)^2 = (\sum_{h}|h|^{-s})(\sum_g|g|^{-s}) = \sum_f(\sum_{hg=f}1)|f|^{-s} = \sum_f d(f)|f|^{-s}=D_d(s) = \sum_{n \mathop =0}^\infty D(n)u^{n}</math>
 
where <math>u=q^{-s}</math>.
 
Expanding the right-hand side into power series we get,
 
: <math>D(n)=(n+1)q^n.</math>
 
Substitute <math>x=q^n</math> the above equation becomes:
: <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]].
 
: <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]].
 
Not much is known about the error term for the integers, while in the polynomials case, there is no error term!
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====Polynomial von Mangoldt function====
The Polynomial [[von Mangoldt function]] is defined by:
<math display="block">\Lambda_{A}(f) = \begin{cases}
\log |P| & \mboxtext{if }f=|P|^k \text{ for some prime monic} P \text{ and integer } k \ge 1, \\
0 & \mboxtext{otherwise.}
\end{cases}</math>
 
Where the logarithm is taken on the basis of ''q''.
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We have,
<math display="block">\begin{align}
 
: <math>
\begin{align}
\sum_{f|m}\Lambda_{A}(f)
&= \sum_{(i_1,\ldots,i_l)|0\le i_j \le e_j} \Lambda_A(\prod_{j \mathop =1}^l P_j^{i_j})=\sum_{j \mathop =1}^l \sum_{i \mathop =1}^{e_i}\Lambda_A (P_j^i) =\sum_{j \mathop =1}^l \sum_{i \mathop =1}^{e_i}\log|P_j|\\
&= \sum_{j \mathop =1}^l e_j\log|P_j| =\sum_{j \mathop =1}^l \log|P_j|^{e_j}=\log|(\prod_{i \mathop =1}^l P_i^{e_i})|\\
&= \log(m)
\end{align}</math>
</math>
 
Hence,
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and we get that,
 
: <math>\zeta_{A}(s)D_{\Lambda_{A}}(s) = \sum_{m}log|m||m|^{-s}.</math>
 
Now,
 
: <math>\sum_m |m|^s = \sum_n \sum_{\deg m = n} u^n=\sum_n q^n u^{n}=\sum_n q^{n(1-s)}.</math>
 
Thus,
 
: <math>\frac{d}{ds}\sum_m |m|^s=-\sum_n \log(q^n)q^{n(1-s)} =-\sum_n \sum_{\deg(f)=n} \log(q^n)q^{-ns}= -\sum_f \log|f||f|^{-s}.</math>
 
We got that:
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Now,
 
: <math>\sum_{m}\Lambda_A (m)|m|^{-s}= \sum_n(\sum_{\deg(m)=n}\Lambda_A(m)q^{-sm}) =\sum_n(\sum_{\deg(m)=n}\Lambda_A(m))u^n=\frac{-\zeta'_A(s)}{\zeta_A(s)}=\frac{q^{1-s}log(q)}{1-q^{1-s}} = \log(q) \sum_{n\mathop=1}^\infty q^n u^n</math>
 
Hence,