Average order of an arithmetic function: Difference between revisions

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Let <math>f</math> be an [[arithmetic function]]. We say that an ''average order'' of <math>f</math> is <math>g</math> if

<math display="block"> \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) </math>

:<math> \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) </math>

as <math>x</math> tends to infinity.

Line 10 ⟶ 8:

In cases where the limit

<math display="block">\lim_{N \to \infty} \frac{1}{N}\sum_{n \le N} f(n)=c</math>

exists, it is said that <math>f</math> has a '''mean value''' ('''average value''') <math>c</math>. If in addition the constant <math>c</math> is not zero, then the constant function <math>g(x)=c</math> is an average order of <math>f</math>.

: <math>\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n \le N} f(n)=c</math>

==Examples==

exists, it is said that <math>f</math> has a '''mean value''' ('''average value''') <math>c</math>.

* An average order of {{math|''d''(''n'')}}, the [[Divisor function|number of divisors]] of {{math|''n''}}, is {{math|log ''n''}};

* An average order of {{math|''σ''(''n'')}}, the [[Divisor function|sum of divisors]] of {{math|''n''}}, is {{math|''n''π²{{thinsp|/}}6}};

==Examples==<!-- &#x200A; is hair space (&hairsp; doesn't work). -->

* An average order of {{math|''dφ''&#x200A;(''n'')}}, the [[DivisorEuler's function|numbertotient of divisorsfunction]] of {{math|''n''}}, is {{math|log 6''n''{{thinsp|/}}π²}};

* An average order of {{math|''σr''&#x200A;(''n'')}}, the [[Divisor function|sumnumber of divisors]]ways of expressing {{math|''n''}} as a sum of two squares, is {{math|''n''&#x200A;π²{{thinsp|/}}12}};

* An average order of {{math|''φ''&#x200A;(''n'')}},representations [[Euler'sof totienta function]]natural number as a sum of {{math|''n''}},three squares is {{math|3&#x200A;4π''n''{{thinsp|/}}π²3}};

* An average ordernumber of {{math|''r''&#x200A;(''n'')}}, the numberdecompositions of waysa ofnatural expressing {{math|''n''}}number asinto a sum of twoone squares,or more consecutive prime numbers is {{math|π''n'' log2}};

* TheAn average order of representations{{math|''ω''(''n'')}}, ofthe a natural[[Prime factors|number as a sum of threedistinct squaresprime isfactors]] of {{math|4π&#x200A;''n''}}, is {{thinspmath|/}}3loglog ''n''}};

* TheAn average numberorder of decompositions{{math|Ω(''n'')}}, of athe natural[[Prime factors|number intoof aprime sumfactors]] of one or more consecutive prime numbers{{math|''n''}}, is {{math|loglog ''n'' log&#x200A;2}};

* AnThe average[[prime ordernumber oftheorem]] {{math|''ω''&#x200A;(''n'')}},is equivalent to the [[Primestatement factors|numberthat ofthe distinct[[von primeMangoldt factorsfunction]] of {{math|Λ(''n'')}}, ishas {{math|log&#x200A;logaverage ''n''}}value 1;

* AnThe average ordervalue of {{math|Ω&#x200A;''μ''(''n'')}}, the [[PrimeMöbius factors|number of prime factorsfunction]] of {{math|''n''}}, is {{math|log&#x200Azero;log ''n''}};this is again equivalent to the [[prime number theorem]].

* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] {{math|Λ&#x200A;(''n'')}} has average order 1;

* An average order of {{math|''μ''&#x200A;(''n'')}}, the [[Möbius function]], is zero; this is again equivalent to the [[prime number theorem]].

==Calculating mean values using Dirichlet series==

In case <math>F</math> is of the form

<math display="block">F(n)=\sum_{d \mid n} f(d),</math>

: <math>F(n)=\sum_{d \mid n} f(d),</math>

for some arithmetic function <math>f(n)</math>, one has,

{{NumBlk||<math display="block">\sum_{n \le x} F(n) = \sum_{d \le x} f(d) \sum_{n\le x, d\mid n} 1

= \sum_{d \le x} f(d)[x/d]

= x\sum_{d \le x} \frac{f(d)}{d} \text{ } + O{\left(\sum_{d \le x} |f(d)|\right)}. </math>|{{EquationRef|1}}}}

Generalized identities of the previous form are found [[Divisor_sum_identities#Average_order_sum_identities|here]]. This identity often provides a practical way to calculate the mean value in terms of the [[Riemann zeta function]]. This is illustrated in the following example.

: <math>\sum_{n \le x} F(n)=\sum_{d \le x} f(d) \sum_{n\le x, d\mid n} 1=\sum_{d \le x} f(d)[x/d] = x\sum_{d \le x} \frac{f(d)}{d} \text{ } + O(\sum_{d \le x} |f(d)|).\qquad\qquad (1)</math>

===The density of the ''k''^th-power-free integers in {{mathbb|N}}===

Generalizations of the previous identity are found [[Divisor_sum_identities#Average_order_sum_identities|here]]. This identity often provides a practical way to calculate the mean value in terms of the [[Riemann zeta function]]. This is illustrated in the following example.

For an integer <math>k \geq 1</math> the set <math>Q_k</math> of '''''k''^th-power-free''' integers is

<math display="block">Q_k :=\{n \in \mathbb{Z}\mid n \text{ is not divisible by } d^k \text{ for any integer } d\ge 2\}.</math>

We calculate the [[natural density]] of these numbers in {{mathbb|N}}, that is, the average value of [[indicator function|<math>1_{Q_k}</math>]], denoted by <math>\delta(n)</math>, in terms of the [[zeta function]].

===The density of the k-th power free integers in {{math|N}}===

For an integer <math>k \geq 1</math> the set <math>Q_k</math> of '''''k''-th-power-free''' integers is

The function <math>\delta</math> is multiplicative, and since it is bounded by 1, its [[Dirichlet series]] converges absolutely in the half-plane <math>\operatorname{Re}(s)>1</math>, and there has [[Euler product]]

: <math>Q_k :=\{n \in \mathbb{Z}\mid n \text{ is not divisible by } d^k \text{ for any integer } d\ge 2\}.</math>

<math display="block">\sum_{Q_k}n^{-s} = \sum_n \delta(n)n^{-s}

= \prod_p \left(1+p^{-s}+\cdots +p^{-s(k-1)}\right)

We calculate the [[natural density]] of these numbers in {{math|'''N'''}}, that is, the average value of [[indicator function|<math>1_{Q_k}</math>]], denoted by <math>\delta(n)</math>, in terms of the [[zeta function]].

= \prod_p \left(\frac{1-p^{-sk}}{1-p^{-s}}\right) = \frac{\zeta(s)}{\zeta(sk)}. </math>

The function <math>\delta</math> is multiplicative, and since it is bounded by 1, its [[Dirichlet series]] converges absolutely in the half-plane <math>\mathrm{Re}(s)<1</math>, and there has [[Euler product]]

: <math>\sum_{Q_k}n^{-s}=\sum_n \delta(n)n^{-s}=\prod_p (1+p^{-s}+\cdots +p^{-s(k-1)})=\prod_p \left(\frac{1-p^{-sk}}{1-p^{-s}}\right)=\frac{\zeta(s)}{\zeta(sk)}. </math>

By the [[Möbius inversion]] formula, we get

<math display="block">\frac{1}{\zeta(ks)} = \sum_{n}\mu(n)n^{-ks},</math>

: <math>

\frac{1}{\zeta(ks)}=\sum_{n}\mu(n)n^{-ks},

</math>

where <math>\mu</math> stands for the [[Möbius function]]. Equivalently,

<math display="block">\frac{1}{\zeta(ks)}=\sum_{n}f(n)n^{-s},</math>

: <math>

\frac{1}{\zeta(ks)}=\sum_{n}f(n)n^{-s},

</math>

where <math>f(n)=\begin{cases}

\;\;\, \mu(d) & n=d^{k}\\

\;\;\, 0 & \text{otherwise},

\end{cases}</math>

and hence,

<math display="block">\frac{\zeta(s)}{\zeta(sk)} = \sum_n \left(\sum_{d\mid n}f(d)\right)n^{-s}.</math>

: <math>\frac{\zeta(s)}{\zeta(sk)}=\sum_n (\sum_{d\mid n}f(d))n^{-s}.</math>

By comparing the coefficients, we get

<math display="block">\delta(n)=\sum_{d\mid n}f(d).</math>

Using {{EquationNote|1|(1)}}, we get

: <math>\delta(n)=\sum_{d\mid n}f(d)n^{-s}.</math>

<math display="block">\sum_{d \le x}\delta(d)=x\sum_{d \le x}(f(d)/d)+O(x^{1/k}).</math>

Using (1), we get

: <math>\sum_{d \le x}\delta(d)=x\sum_{d \le x}(f(d)/d)+O(x^{1/k}).</math>

We conclude that,

<math display="block"> \sum_{\stackrel{n\in Q_k}{n \le x}} 1 = \frac{x}{\zeta(k)}+O(x^{1/k}), </math>

: <math>

\sum_{n\in Q_k, n \le x}1=\frac{x}{\zeta(k)}+O(x^{1/k}),

</math>

where for this we used the relation

<math display="block">\sum_n (f(n)/n)=\sum_n f(n^k)n^{-k}=\sum_n \mu(n)n^{-k}=\frac{1}{\zeta(k)},</math>

: <math>\sum_n (f(n)/n)=\sum_n f(n^k)n^{-k}=\sum_n \mu(n)n^{-k}=\frac{1}{\zeta(k)},</math>

which follows from the Möbius inversion formula.

In particular, the density of the [[square-free integers]] is <math display="inline">\zeta(2)^{-1}=\frac{6}{\pi^{2}}</math>.

===Visibility of lattice points===

We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them.

Now, if {{math|1=gcd(''a'', ''b'') = ''d'' > 1}}, then writing ''a'' = ''da''², ''b'' = ''db''² one observes that the point (''a''², ''b''²) is on the line segment which joins (0, 0) to (''a'', ''b'') and hence (''a'', ''b'') is not visible from the origin. Thus (''a'', ''b'') is visible from the origin implies that (''a'', ''b'') = 1. Conversely, it is also easy to see that gcd(''a'', ''b'') = 1 implies that there is no other [[integer lattice point]] in the segment joining (0, 0) to (''a'', ''b'').

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

(''a''², ''b''²) is on the line segment which joins (0,0) to (''a'', ''b'') and hence (''a'', ''b'') is not visible from the origin. Thus (''a'', ''b'') is visible from the origin implies that (''a'', ''b'') = 1. Conversely, it is also easy to see that gcd(''a'', ''b'') = 1 implies that there is no other integer lattice point in the segment joining (0,0) to (''a'',''b'').

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,

<math display="block">\lim_{N\to\infty} \frac{1}{N}\sum_{n\le N} \frac{\varphi(n)}{n} = \frac{6}{\pi^2}=\frac{1}{\zeta(2)}. </math>

<math display="inline">\frac{1}{\zeta(2)}</math> is also the natural density of the square-free numbers in {{mathbb|N}}. In fact, this is not a coincidence. Consider the ''k''-dimensional lattice, <math>\mathbb{Z}^{k}</math>. The natural density of the points which are visible from the origin is <math display="inline">\frac{1}{\zeta(k)}</math>, which is also the natural density of the ''k''-th free integers in {{mathbb|N}}.

: <math>\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{n\le N} \frac{\varphi(n)}{n} = \frac{6}{\pi^2}=\frac{1}{\zeta(2)}. </math>

interestingly, <math>\frac{1}{\zeta(2)}</math> is also the natural density of the square-free numbers in {{math|'''N'''}}. In fact, this is not a coincidence. Consider the ''k''-dimensional lattice, <math>\mathbb{Z}^{k}</math>. The natural density of the points which are visible from the origin is <math>\frac{1}{\zeta(k)}</math>, which is also the natural density of the ''k''-th free integers in {{math|'''N'''}}.

===Divisor functions===

Consider the generalization of <math>d(n)</math>:

<math display="block">\sigma_\alpha(n)=\sum_{d\mid n}d^\alpha.</math>

: <math>\sigma_\alpha(n)=\sum_{d\mid n}d^\alpha.</math>

The following are true:

<math display="block">

: <math>

\sum_{n\le x}\sigma_{\alpha}(n)=

\begin{cases}

\;\;\sum_{n\le x}\sigma_\alpha(n)=\frac{\zeta(\alpha+1)}{\alpha+1}x^{\alpha+1}+O(x^\beta) & \text{if } \alpha>0,\alpha \ne 1, \\

\;\;\sum_{n\le x}\sigma_{1}(n)=\frac{\zeta(2)}{2}x^2+O(x \log x) & \text{if } \alpha=1, \\

\;\;\sum_{n\le x}\sigma_{-1}(n)=\zeta(2)x+O(\log x) & \text{if } \alpha=-1, \\

\;\;\sum_{n\le x}\sigma_\alpha(n)=\zeta(-\alpha+1)x+O(x^{\max(0,1+\alpha)}) & \text{otherwise.}

\end{cases}

</math>

where <math>\beta=\max(1,\alpha)</math>.

where <math>\beta=max(1,\alpha)</math>.

==Better average order==

This notion is best discussed through an example. From

:<math display="block"> \sum_{n\le x}d(n)=x\log x+(2\gamma-1)x+o(x)</math>

(<math>\gamma</math> is the [[Euler–Mascheroni constant]]) and

:<math display="block"> \sum_{n\le x}\log n=x\log x-x+O(\log x),</math>

we have the asymptotic relation

:<math display="block">\sum_{n\le x}(d(n)-(\log n+2\gamma))=o(x)\quad(x\rightarrowto\infty),</math>

which suggests that the function <math>\log n+2\gamma</math> is a better choice of average order for <math>d(n)</math> than simply <math>\log n</math>.

==Mean values over {{math|F_q[''x'']}}==

===Definition===

Let ''h''(''x'') be a function on the set of [[monic polynomial]]s over [[finite field|'''F_q''']]. For <math>n\ge 1</math> we define

<math display="block">\text{Ave}_n(h)=\frac{1}{q^n}\sum_{f \text{ monic},\deg(f)= n}h(f).</math>

: <math>\text{Ave}_n(h)=\frac{1}{q^n}\sum_{f \text{ monic},\deg(f)= n}h(f).</math>

This is the mean value (average value) of ''h'' on the set of monic polynomials of degree ''n''. We say that ''g''(''n'') is an '''average order''' of ''h'' if

<math display="block">\text{Ave}_n(h) \sim g(n)</math>

: <math>\text{Ave}_n(h) \sim g(n)</math>

as ''n'' tends to infinity.

In cases where the limit,

<math display="block">\lim_{n\to\infty}\text{Ave}_n(h) = c</math>

: <math>\lim_{n\rightarrow\infty}\text{Ave}_n(h)=c</math>

exists, it is said that ''h'' has a '''mean value''' ('''average value''') ''c''.

===Zeta function and Dirichlet series in {{math|F_q[X]}}===

Let {{math|1='''F_q'''['''X]'''}}] = ''A''}} be the [[ring of polynomials]] over the [[finite field]] {{math|'''F_q'''}}.

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 display="block">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.

: <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.

The polynomial zeta function is then

<math display="block">\zeta_{A}(s)=\sum_{f\text{ monic}} |f|^{-s}.</math>

: <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 display="block">D_{h}(s) = \prod_{P} \left(\sum_{n=0}^{\infty} h(P^{n}) \left|P\right|^{-sn}\right),</math>

where the product runs over all monic irreducible polynomials ''P''.

For example, the product representation of the zeta function is as for the integers: <math display="inline">\zeta_{A}(s) = \prod_{P} \left(1 - \left|P\right|^{-s}\right)^{-1}</math>.

: <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''.

For example, the product representation of the zeta function is as for the integers: <math>\zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}</math>.

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

In a similar way, If ''f'' and ''g'' are two polynomial arithmetic functions, one defines ''f''&nbsp;*&nbsp;''g'', the ''Dirichlet convolution'' of ''f'' 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 display="block">\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.

Line 198 ⟶ 158:

By multiplicativity of <math>\delta</math>:

<math display="block">\sum_f \frac{\delta(f)}{|f|^s} = \prod_{P} \left(\sum_{j=0}^{k-1}(|P|^{-js})\right) = \prod_{P}\frac{1-|P|^{-sk}}{1-|P|^{-s}} = \frac{\zeta_A(s)}{\zeta_A(sk)} = \frac{1-q^{1-ks}}{1-q^{1-s}} = \frac{\zeta_A(s)}{\zeta_A(ks)}</math>

: <math>\sum_f \frac{\delta(f)}{|f|^s}=\prod_{P}(\sum_{j \mathop =0}^{k-1}(|P|^{-js}))=\prod_{P}\frac{1-|P|^{-sk}}{1-|P|^{-s}}=\frac{\zeta_A(s)}{\zeta_A(sk)}=\frac{1-q^{1-ks}}{1-q^{1-s}}=\frac{\zeta_A(s)}{\zeta_A(ks)}</math>

Denote <math>b_n</math> the number of ''k''-th power monic polynomials of degree ''n'', we get

<math display="block">\sum_{f}\frac{\delta(f)}{|f|^{s}}=\sum_{n}\sum_{\text{def}f=n}\delta(f)|f|^{-s}=\sum_{n}b_{n}q^{-sn}. </math>

: <math>\sum_{f}\frac{\delta(f)}{|f|^{s}}=\sum_{n}\sum_{\text{def}f=n}\delta(f)|f|^{-s}=\sum_{n}b_{n}q^{-sn}. </math>

Making the substitution <math>u=q^{-s}</math> we get:

<math display="block">\frac{1-qu^k}{1-qu}=\sum_{n=0}^\infty b_{n}u^{n}.</math>

: <math>\frac{1-qu^k}{1-qu}=\sum_{n \mathop =0}^\infty b_{n}u^{n}.</math>

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 \\

<math>b_{n}=\begin{cases}

q^{n}(1-q^{1-k}) &\text{otherwise}

\;\;\,q^{n} & n\le k-1 \\

\;\;\, q^{n}(1-q^{1-k}) &\text{otherwise}

\end{cases}</math>

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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====Polynomial Divisor functions====

In {{math|'''F_q[X]'''}}, we define

<math display="block">\sigma _{k}(m)=\sum_{f|m, \text{ monic}}|f|^k.</math>

: <math>\sigma _{k}(m)=\sum_{f|m, \text{ monic}}|f|^k.</math>

We will compute <math>\text{Ave}_{n}(\sigma_{k})</math> for <math>k\ge 1</math>.

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 display="block">\sum_{m}\sigma_{k}(m)|m|^{-s}=\zeta_{A}(s)\sum_{m}h(m)|m|^{-s}.</math>

: <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 display="block">\text{LHS}=\sum_n\left(\sum_{\deg(m)=n} \sigma_k(m)\right)u^n,</math>and by [[Cauchy product]] we get,

<math display="block">\begin{align}

: <math>\text{LHS}=\sum_n(\sum_{\deg(m)=n} \sigma_k(m))u^n</math>, and by [[Cauchy product]] we get,

: <math>

\begin{align}

\text{RHS}

&=\sum_n q^{n(1-s)}\sum_{n}\left(\sum_{\deg(m)=n}h(m)\right)u^n \\

&=\sum_n q^n u^n \sum_{l}q^l q^{lk}u^l \\

&=\sum_n \left(\sum_{j \mathop =0}^{n}q^{n-j}q^{jk+j}\right) \\

&=\sum_n \left(q^n\left(\frac{1-q^{k(n+1)}}{1-q^k}\right)\right) u^n.

\end{align}</math>

</math>

Finally we get that,

<math display="block"> \text{Ave}_n\sigma_k=\frac{1-q^{k(n+1)}}{1-q^k}. </math>

: <math> \text{Ave}_n\sigma_k=\frac{1-q^{k(n+1)}}{1-q^k}. </math>

Notice that

<math display="block">q^n \text{Ave}_n\sigma_{k} = q^{n(k+1)}\left(\frac{1-q^{-k(n+1)}}{1-q^{-k}}\right) = q^{n(k+1)}\left(\frac{\zeta(k+1)}{\zeta(kn+k+1)}\right)</math>

: <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

<math display="block">\sum_{\deg(m)=n, m \text{ monic}} \sigma_k(m) = x^{k+1}\left(\frac{\zeta(k+1)}{\zeta(kn+k+1)}\right)</math>

: <math>\sum_{\deg(m)=n, m \text{ monic}} \sigma_k(m)=x^{k+1}(\frac{\zeta(k+1)}{\zeta(kn+k+1)})</math>

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====

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

<math>= \zeta_A(s)^2=left(\sum_{h}|h|^{-s}\right)\left(\sum_g|g|^{-s}\right)
= \sum_f\left(\sum_{hg=f} 1 \right)|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 display="block">D(n)=(n+1)q^n.</math>

: <math>D(n)=(n+1)q^n.</math>

Substitute <math>x=q^n</math> the above equation becomes:

<math display="block">D(n)=x \log_q(x)+x</math> which resembles closely the analogous result for integers <math display="inline">\sum_{k=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. This is because of the very simple nature of the zeta function <math>\zeta_{A}(s)</math>, and that it has no zeros.

: <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.

====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''.

Where the logarithm is taken on the basis of ''q''.

'''Proposition.''' The mean value of <math>\Lambda_{A}</math> is exactly ''1''.

'''Proof.'''

Let ''m'' be a monic polynomial, and let <math display="inline">m = \prod_{i \mathop =1}^{l} P_{i}^{e_i}</math> be the prime decomposition of ''m''.

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\left(\prod_{j \mathop =1}^l P_j^{i_j}\right)
= \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\left|\left(\prod_{i \mathop =1}^l P_i^{e_i}\right)\right|\\

&= \log(m)

\end{align}</math>

</math>

Hence,

<math display="block">\mathbb{I}\cdot\Lambda_A(m)=\log|m|</math>

: <math>\mathbb{I}\cdot\Lambda_A(m)=\log|m|</math>

and we get that,

<math display="block">\zeta_{A}(s)D_{\Lambda_{A}}(s) = \sum_{m}\log\left|m\right|\left|m\right|^{-s}.</math>

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

Now,

<math display="block">\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>

: <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 display="block">\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\left|f\right|\left|f\right|^{-s}.</math>

: <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:

<math display="block">D_{\Lambda_A}(s) = \frac{-\zeta'_A(s)}{\zeta_{A}(s)}</math>

: <math>D_{\Lambda_A}(s)=\frac{-\zeta'_A(s)}{\zeta_{A}(s)}</math>

Now,

<math display="block">\begin{align}

: <math>\sum_{m}\Lambda_A (m)|m|^{-s}
&= \sum_n \left(\sum_{\deg(m)=n}\Lambda_A(m)q^{-sm}\right)
= \sum_n \left(\sum_{\deg(m)=n} \Lambda_A(m)\right)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>

\end{align}</math>

Hence,

<math display="block">\sum_{\deg(m)=n}\Lambda_A (m)=q^n \log(q),</math>

: <math>\sum_{\deg(m)=n}\Lambda_A (m)=q^n \log(q),</math>

and by dividing by <math>q^n</math> we get that,

<math display="block">\text{Ave}_n \Lambda_A(m)=\log(q)=1. </math>

: <math>\text{Ave}_n \Lambda_A(m)=\log(q)=1. </math>

====Polynomial Euler totient function====

Define [[Euler totient function]] polynomial analogue, <math>\Phi</math>, to be the number of elements in the group <math>(A/fA)^{*}</math>. We have,

<math display="block">\sum_{\deg f=n, f\text{ monic}}\Phi(f)=q^{2n}(1-q^{-1}).</math>

: <math>\sum_{\deg f=n, f\text{ monic}}\Phi(f)=q^{2n}(1-q^{-1}).</math>

==See also==

Line 351 ⟶ 290:

* [[Normal order of an arithmetic function]]

* [[Extremal orders of an arithmetic function]]

* [[Divisor_sum_identitiesDivisor sum identities#Average_order_sum_identitiesAverage order sum identities|Divisor sum identities]]

==References==

* {{Hardy and Wright|citation=cite book}} Pppp.&nbsp;347–360

* {{cite book | title=Introduction to Analytic and Probabilistic Number Theory | author=Gérald Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-41261-7 | zbl=0831.11001 | pages=36–55 }}

* {{Citation

Line 362 ⟶ 301:

|year=1976

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

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

|url-access=registration

|url=https://archive.org/details/introductiontoan00apos_0

}}

* {{Citation

|title=Number Theory in Function Fields