Examples
edit
An average order of d (n )number of divisors of n log n ;
An average order of σ (n )sum of divisors of n n π2 / 6
An average order of φ (n )Euler's totient function of n 6n / π2 ;
An average order of r (n )n π ;
The average order of representations of a natural number as a sum of three squares is 4πn / 3 ;
The average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is n log2
An average order of ω (n )number of distinct prime factors of n loglog n ;
An average order of Ω(n ) , the number of prime factors of n loglog n ;
The prime number theorem is equivalent to the statement that the von Mangoldt function Λ(n ) has average order 1;
An average value of μ (n )Möbius function , is zero; this is again equivalent to the prime number theorem . Calculating mean values using Dirichlet series
edit
In case
F
{\displaystyle F}
F
(
n
)
=
∑
d
∣
n
f
(
d
)
,
{\displaystyle F(n)=\sum _{d\mid n}f(d),}
f
(
n
)
{\displaystyle f(n)}
∑
n
≤
x
F
(
n
)
=
∑
d
≤
x
f
(
d
)
∑
n
≤
x
,
d
∣
n
1
=
∑
d
≤
x
f
(
d
)
[
x
/
d
]
=
x
∑
d
≤
x
f
(
d
)
d
+
O
(
∑
d
≤
x
|
f
(
d
)
|
)
.
{\displaystyle \sum _{n\leq x}F(n)=\sum _{d\leq x}f(d)\sum _{n\leq x,d\mid n}1=\sum _{d\leq x}f(d)[x/d]=x\sum _{d\leq x}{\frac {f(d)}{d}}{\text{ }}+O{\left(\sum _{d\leq x}|f(d)|\right)}.}
(1 )
Generalized identities of the previous form are found 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.
The density of the k th -power-free integers in ℕ
edit
For an integer
k
≥
1
{\displaystyle k\geq 1}
Q
k
{\displaystyle Q_{k}}
k th -power-free
Q
k
:=
{
n
∈
Z
∣
n
is not divisible by
d
k
for any integer
d
≥
2
}
.
{\displaystyle Q_{k}:=\{n\in \mathbb {Z} \mid n{\text{ is not divisible by }}d^{k}{\text{ for any integer }}d\geq 2\}.}
We calculate the natural density of these numbers in ℕ, that is, the average value of
1
Q
k
{\displaystyle 1_{Q_{k}}}
δ
(
n
)
{\displaystyle \delta (n)}
zeta function .
The function
δ
{\displaystyle \delta }
Dirichlet series converges absolutely in the half-plane
Re
(
s
)
>
1
{\displaystyle \operatorname {Re} (s)>1}
Euler product
∑
Q
k
n
−
s
=
∑
n
δ
(
n
)
n
−
s
=
∏
p
(
1
+
p
−
s
+
⋯
+
p
−
s
(
k
−
1
)
)
=
∏
p
(
1
−
p
−
s
k
1
−
p
−
s
)
=
ζ
(
s
)
ζ
(
s
k
)
.
{\displaystyle \sum _{Q_{k}}n^{-s}=\sum _{n}\delta (n)n^{-s}=\prod _{p}\left(1+p^{-s}+\cdots +p^{-s(k-1)}\right)=\prod _{p}\left({\frac {1-p^{-sk}}{1-p^{-s}}}\right)={\frac {\zeta (s)}{\zeta (sk)}}.}
By the Möbius inversion formula, we get
1
ζ
(
k
s
)
=
∑
n
μ
(
n
)
n
−
k
s
,
{\displaystyle {\frac {1}{\zeta (ks)}}=\sum _{n}\mu (n)n^{-ks},}
μ
{\displaystyle \mu }
Möbius function . Equivalently,
1
ζ
(
k
s
)
=
∑
n
f
(
n
)
n
−
s
,
{\displaystyle {\frac {1}{\zeta (ks)}}=\sum _{n}f(n)n^{-s},}
f
(
n
)
=
{
μ
(
d
)
n
=
d
k
0
otherwise
,
{\displaystyle f(n)={\begin{cases}\mu (d)&n=d^{k}\\0&{\text{otherwise}},\end{cases}}}
ζ
(
s
)
ζ
(
s
k
)
=
∑
n
(
∑
d
∣
n
f
(
d
)
)
n
−
s
.
{\displaystyle {\frac {\zeta (s)}{\zeta (sk)}}=\sum _{n}\left(\sum _{d\mid n}f(d)\right)n^{-s}.}
By comparing the coefficients, we get
δ
(
n
)
=
∑
d
∣
n
f
(
d
)
.
{\displaystyle \delta (n)=\sum _{d\mid n}f(d).}
Using (1)
∑
d
≤
x
δ
(
d
)
=
x
∑
d
≤
x
(
f
(
d
)
/
d
)
+
O
(
x
1
/
k
)
.
{\displaystyle \sum _{d\leq x}\delta (d)=x\sum _{d\leq x}(f(d)/d)+O(x^{1/k}).}
We conclude that,
∑
n
≤
x
n
∈
Q
k
1
=
x
ζ
(
k
)
+
O
(
x
1
/
k
)
,
{\displaystyle \sum _{\stackrel {n\in Q_{k}}{n\leq x}}1={\frac {x}{\zeta (k)}}+O(x^{1/k}),}
∑
n
(
f
(
n
)
/
n
)
=
∑
n
f
(
n
k
)
n
−
k
=
∑
n
μ
(
n
)
n
−
k
=
1
ζ
(
k
)
,
{\displaystyle \sum _{n}(f(n)/n)=\sum _{n}f(n^{k})n^{-k}=\sum _{n}\mu (n)n^{-k}={\frac {1}{\zeta (k)}},}
In particular, the density of the square-free integers is
ζ
(
2
)
−
1
=
6
π
2
{\textstyle \zeta (2)^{-1}={\frac {6}{\pi ^{2}}}}
Visibility of lattice points
edit
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 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.
Notice that
φ
(
n
)
n
{\displaystyle {\frac {\varphi (n)}{n}}}
{
(
r
,
s
)
∈
N
:
max
(
|
r
|
,
|
s
|
)
=
n
}
{\displaystyle \{(r,s)\in \mathbb {N} :\max(|r|,|s|)=n\}}
Thus, one can show that the natural density of the points which are visible from the origin is given by the average,
lim
N
→
∞
1
N
∑
n
≤
N
φ
(
n
)
n
=
6
π
2
=
1
ζ
(
2
)
.
{\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\sum _{n\leq N}{\frac {\varphi (n)}{n}}={\frac {6}{\pi ^{2}}}={\frac {1}{\zeta (2)}}.}
1
ζ
(
2
)
{\textstyle {\frac {1}{\zeta (2)}}}
k -dimensional lattice,
Z
k
{\displaystyle \mathbb {Z} ^{k}}
1
ζ
(
k
)
{\textstyle {\frac {1}{\zeta (k)}}}
k -th free integers in ℕ.
Divisor functions
edit
Consider the generalization of
d
(
n
)
{\displaystyle d(n)}
σ
α
(
n
)
=
∑
d
∣
n
d
α
.
{\displaystyle \sigma _{\alpha }(n)=\sum _{d\mid n}d^{\alpha }.}
The following are true:
∑
n
≤
x
σ
α
(
n
)
=
{
∑
n
≤
x
σ
α
(
n
)
=
ζ
(
α
+
1
)
α
+
1
x
α
+
1
+
O
(
x
β
)
if
α
>
0
,
∑
n
≤
x
σ
−
1
(
n
)
=
ζ
(
2
)
x
+
O
(
log
x
)
if
α
=
−
1
,
∑
n
≤
x
σ
α
(
n
)
=
ζ
(
−
α
+
1
)
x
+
O
(
x
max
(
0
,
1
+
α
)
)
otherwise.
{\displaystyle \sum _{n\leq x}\sigma _{\alpha }(n)={\begin{cases}\;\;\sum _{n\leq x}\sigma _{\alpha }(n)={\frac {\zeta (\alpha +1)}{\alpha +1}}x^{\alpha +1}+O(x^{\beta })&{\text{if }}\alpha >0,\\\;\;\sum _{n\leq x}\sigma _{-1}(n)=\zeta (2)x+O(\log x)&{\text{if }}\alpha =-1,\\\;\;\sum _{n\leq x}\sigma _{\alpha }(n)=\zeta (-\alpha +1)x+O(x^{\max(0,1+\alpha )})&{\text{otherwise.}}\end{cases}}}
β
=
max
(
1
,
α
)
{\displaystyle \beta =\max(1,\alpha )}
Better average order
edit
Mean values over Fq [x ]
edit
Definition
edit
Let h (x ) be a function on the set of monic polynomials over Fq
n
≥
1
{\displaystyle n\geq 1}
Ave
n
(
h
)
=
1
q
n
∑
f
monic
,
deg
(
f
)
=
n
h
(
f
)
.
{\displaystyle {\text{Ave}}_{n}(h)={\frac {1}{q^{n}}}\sum _{f{\text{ monic}},\deg(f)=n}h(f).}
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
Ave
n
(
h
)
∼
g
(
n
)
{\displaystyle {\text{Ave}}_{n}(h)\sim g(n)}
n tends to infinity.
In cases where the limit,
lim
n
→
∞
Ave
n
(
h
)
=
c
{\displaystyle \lim _{n\to \infty }{\text{Ave}}_{n}(h)=c}
h has a mean value (average value ) c .
Zeta function and Dirichlet series in Fq [X]
edit
Let Fq [X ] = A ring of polynomials over the finite field Fq
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
D
h
(
s
)
=
∑
f
monic
h
(
f
)
|
f
|
−
s
,
{\displaystyle D_{h}(s)=\sum _{f{\text{ monic}}}h(f)|f|^{-s},}
g
∈
A
{\displaystyle g\in A}
|
g
|
=
q
deg
(
g
)
{\displaystyle |g|=q^{\deg(g)}}
g
≠
0
{\displaystyle g\neq 0}
|
g
|
=
0
{\displaystyle |g|=0}
The polynomial zeta function is then
ζ
A
(
s
)
=
∑
f
monic
|
f
|
−
s
.
{\displaystyle \zeta _{A}(s)=\sum _{f{\text{ monic}}}|f|^{-s}.}
Similar to the situation in N multiplicative function h has a product representation (Euler product):
D
h
(
s
)
=
∏
P
(
∑
n
=
0
∞
h
(
P
n
)
|
P
|
−
s
n
)
,
{\displaystyle D_{h}(s)=\prod _{P}\left(\sum _{n=0}^{\infty }h(P^{n})\left|P\right|^{-sn}\right),}
P .
For example, the product representation of the zeta function is as for the integers:
ζ
A
(
s
)
=
∏
P
(
1
−
|
P
|
−
s
)
−
1
{\textstyle \zeta _{A}(s)=\prod _{P}\left(1-\left|P\right|^{-s}\right)^{-1}}
Unlike the classical zeta function ,
ζ
A
(
s
)
{\displaystyle \zeta _{A}(s)}
ζ
A
(
s
)
=
∑
f
(
|
f
|
−
s
)
=
∑
n
∑
deg
(
f
)
=
n
q
−
s
n
=
∑
n
(
q
n
−
s
n
)
=
(
1
−
q
1
−
s
)
−
1
.
{\displaystyle \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}.}
In a similar way, If f and g are two polynomial arithmetic functions, one defines f * g , the Dirichlet convolution of f and g , by
(
f
∗
g
)
(
m
)
=
∑
d
∣
m
f
(
m
)
g
(
m
d
)
=
∑
a
b
=
m
f
(
a
)
g
(
b
)
{\displaystyle {\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(m)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b)\end{aligned}}}
divisors d of m , or equivalently over all pairs (a , b ) of monic polynomials whose product is m . The identity
D
h
D
g
=
D
h
∗
g
{\displaystyle D_{h}D_{g}=D_{h*g}}
Examples
edit
The density of the k -th power free polynomials in Fq [X]
edit
Define
δ
(
f
)
{\displaystyle \delta (f)}
f
{\displaystyle f}
k -th power free and 0 otherwise.
We calculate the average value of
δ
{\displaystyle \delta }
k -th power free polynomials in F q
By multiplicativity of
δ
{\displaystyle \delta }
∑
f
δ
(
f
)
|
f
|
s
=
∏
P
(
∑
j
=
0
k
−
1
(
|
P
|
−
j
s
)
)
=
∏
P
1
−
|
P
|
−
s
k
1
−
|
P
|
−
s
=
ζ
A
(
s
)
ζ
A
(
s
k
)
=
1
−
q
1
−
k
s
1
−
q
1
−
s
=
ζ
A
(
s
)
ζ
A
(
k
s
)
{\displaystyle \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)}}}
Denote
b
n
{\displaystyle b_{n}}
k -th power monic polynomials of degree n , we get
∑
f
δ
(
f
)
|
f
|
s
=
∑
n
∑
def
f
=
n
δ
(
f
)
|
f
|
−
s
=
∑
n
b
n
q
−
s
n
.
{\displaystyle \sum _{f}{\frac {\delta (f)}{|f|^{s}}}=\sum _{n}\sum _{{\text{def}}f=n}\delta (f)|f|^{-s}=\sum _{n}b_{n}q^{-sn}.}
Making the substitution
u
=
q
−
s
{\displaystyle u=q^{-s}}
1
−
q
u
k
1
−
q
u
=
∑
n
=
0
∞
b
n
u
n
.
{\displaystyle {\frac {1-qu^{k}}{1-qu}}=\sum _{n=0}^{\infty }b_{n}u^{n}.}
Finally, expand the left-hand side in a geometric series and compare the coefficients on
u
n
{\displaystyle u^{n}}
b
n
=
{
q
n
n
≤
k
−
1
q
n
(
1
−
q
1
−
k
)
otherwise
{\displaystyle b_{n}={\begin{cases}q^{n}&n\leq k-1\\q^{n}(1-q^{1-k})&{\text{otherwise}}\end{cases}}}
Hence,
Ave
n
(
δ
)
=
1
−
q
1
−
k
=
1
ζ
A
(
k
)
{\displaystyle {\text{Ave}}_{n}(\delta )=1-q^{1-k}={\frac {1}{\zeta _{A}(k)}}}
And since it doesn't depend on n this is also the mean value of
δ
(
f
)
{\displaystyle \delta (f)}
Polynomial Divisor functions
edit
In Fq [X]
σ
k
(
m
)
=
∑
f
|
m
,
monic
|
f
|
k
.
{\displaystyle \sigma _{k}(m)=\sum _{f|m,{\text{ monic}}}|f|^{k}.}
We will compute
Ave
n
(
σ
k
)
{\displaystyle {\text{Ave}}_{n}(\sigma _{k})}
k
≥
1
{\displaystyle k\geq 1}
First, notice that
σ
k
(
m
)
=
h
∗
I
(
m
)
{\displaystyle \sigma _{k}(m)=h*\mathbb {I} (m)}
h
(
f
)
=
|
f
|
k
{\displaystyle h(f)=|f|^{k}}
I
(
f
)
=
1
∀
f
{\displaystyle \mathbb {I} (f)=1\;\;\forall {f}}
Therefore,
∑
m
σ
k
(
m
)
|
m
|
−
s
=
ζ
A
(
s
)
∑
m
h
(
m
)
|
m
|
−
s
.
{\displaystyle \sum _{m}\sigma _{k}(m)|m|^{-s}=\zeta _{A}(s)\sum _{m}h(m)|m|^{-s}.}
Substitute
q
−
s
=
u
{\displaystyle q^{-s}=u}
LHS
=
∑
n
(
∑
deg
(
m
)
=
n
σ
k
(
m
)
)
u
n
,
{\displaystyle {\text{LHS}}=\sum _{n}\left(\sum _{\deg(m)=n}\sigma _{k}(m)\right)u^{n},}
Cauchy product we get,
RHS
=
∑
n
q
n
(
1
−
s
)
∑
n
(
∑
deg
(
m
)
=
n
h
(
m
)
)
u
n
=
∑
n
q
n
u
n
∑
l
q
l
q
l
k
u
l
=
∑
n
(
∑
j
=
0
n
q
n
−
j
q
j
k
+
j
)
=
∑
n
(
q
n
(
1
−
q
k
(
n
+
1
)
1
−
q
k
)
)
u
n
.
{\displaystyle {\begin{aligned}{\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=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{aligned}}}
Finally we get that,
Ave
n
σ
k
=
1
−
q
k
(
n
+
1
)
1
−
q
k
.
{\displaystyle {\text{Ave}}_{n}\sigma _{k}={\frac {1-q^{k(n+1)}}{1-q^{k}}}.}
Notice that
q
n
Ave
n
σ
k
=
q
n
(
k
+
1
)
(
1
−
q
−
k
(
n
+
1
)
1
−
q
−
k
)
=
q
n
(
k
+
1
)
(
ζ
(
k
+
1
)
ζ
(
k
n
+
k
+
1
)
)
{\displaystyle 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)}
Thus, if we set
x
=
q
n
{\displaystyle x=q^{n}}
∑
deg
(
m
)
=
n
,
m
monic
σ
k
(
m
)
=
x
k
+
1
(
ζ
(
k
+
1
)
ζ
(
k
n
+
k
+
1
)
)
{\displaystyle \sum _{\deg(m)=n,m{\text{ monic}}}\sigma _{k}(m)=x^{k+1}\left({\frac {\zeta (k+1)}{\zeta (kn+k+1)}}\right)}
∑
n
≤
x
σ
k
(
n
)
=
ζ
(
k
+
1
)
k
+
1
x
k
+
1
+
O
(
x
k
)
{\displaystyle \sum _{n\leq x}\sigma _{k}(n)={\frac {\zeta (k+1)}{k+1}}x^{k+1}+O(x^{k})}
Number of divisors
edit
Let
d
(
f
)
{\displaystyle d(f)}
f and let
D
(
n
)
{\displaystyle D(n)}
d
(
f
)
{\displaystyle d(f)}
ζ
A
(
s
)
2
=
(
∑
h
|
h
|
−
s
)
(
∑
g
|
g
|
−
s
)
=
∑
f
(
∑
h
g
=
f
1
)
|
f
|
−
s
=
∑
f
d
(
f
)
|
f
|
−
s
=
D
d
(
s
)
=
∑
n
=
0
∞
D
(
n
)
u
n
{\displaystyle \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=0}^{\infty }D(n)u^{n}}
u
=
q
−
s
{\displaystyle u=q^{-s}}
Expanding the right-hand side into power series we get,
D
(
n
)
=
(
n
+
1
)
q
n
.
{\displaystyle D(n)=(n+1)q^{n}.}
Substitute
x
=
q
n
{\displaystyle x=q^{n}}
D
(
n
)
=
x
log
q
(
x
)
+
x
{\displaystyle D(n)=x\log _{q}(x)+x}
∑
k
=
1
n
d
(
k
)
=
x
log
x
+
(
2
γ
−
1
)
x
+
O
(
x
)
{\textstyle \sum _{k=1}^{n}d(k)=x\log x+(2\gamma -1)x+O({\sqrt {x}})}
γ
{\displaystyle \gamma }
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
ζ
A
(
s
)
{\displaystyle \zeta _{A}(s)}
Polynomial von Mangoldt function
edit
The Polynomial von Mangoldt function is defined by:
Λ
A
(
f
)
=
{
log
|
P
|
if
f
=
|
P
|
k
for some prime monic
P
and integer
k
≥
1
,
0
otherwise.
{\displaystyle \Lambda _{A}(f)={\begin{cases}\log |P|&{\text{if }}f=|P|^{k}{\text{ for some prime monic}}P{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}}}
q .
Proposition. The mean value of
Λ
A
{\displaystyle \Lambda _{A}}
1 .
Proof.
Let m be a monic polynomial, and let
m
=
∏
i
=
1
l
P
i
e
i
{\textstyle m=\prod _{i=1}^{l}P_{i}^{e_{i}}}
m .
We have,
∑
f
|
m
Λ
A
(
f
)
=
∑
(
i
1
,
…
,
i
l
)
|
0
≤
i
j
≤
e
j
Λ
A
(
∏
j
=
1
l
P
j
i
j
)
=
∑
j
=
1
l
∑
i
=
1
e
i
Λ
A
(
P
j
i
)
=
∑
j
=
1
l
∑
i
=
1
e
i
log
|
P
j
|
=
∑
j
=
1
l
e
j
log
|
P
j
|
=
∑
j
=
1
l
log
|
P
j
|
e
j
=
log
|
(
∏
i
=
1
l
P
i
e
i
)
|
=
log
(
m
)
{\displaystyle {\begin{aligned}\sum _{f|m}\Lambda _{A}(f)&=\sum _{(i_{1},\ldots ,i_{l})|0\leq i_{j}\leq e_{j}}\Lambda _{A}\left(\prod _{j=1}^{l}P_{j}^{i_{j}}\right)=\sum _{j=1}^{l}\sum _{i=1}^{e_{i}}\Lambda _{A}(P_{j}^{i})\\&=\sum _{j=1}^{l}\sum _{i=1}^{e_{i}}\log |P_{j}|\\&=\sum _{j=1}^{l}e_{j}\log |P_{j}|=\sum _{j=1}^{l}\log |P_{j}|^{e_{j}}\\&=\log \left|\left(\prod _{i=1}^{l}P_{i}^{e_{i}}\right)\right|\\&=\log(m)\end{aligned}}}
Hence,
I
⋅
Λ
A
(
m
)
=
log
|
m
|
{\displaystyle \mathbb {I} \cdot \Lambda _{A}(m)=\log |m|}
ζ
A
(
s
)
D
Λ
A
(
s
)
=
∑
m
log
|
m
|
|
m
|
−
s
.
{\displaystyle \zeta _{A}(s)D_{\Lambda _{A}}(s)=\sum _{m}\log \left|m\right|\left|m\right|^{-s}.}
Now,
∑
m
|
m
|
s
=
∑
n
∑
deg
m
=
n
u
n
=
∑
n
q
n
u
n
=
∑
n
q
n
(
1
−
s
)
.
{\displaystyle \sum _{m}|m|^{s}=\sum _{n}\sum _{\deg m=n}u^{n}=\sum _{n}q^{n}u^{n}=\sum _{n}q^{n(1-s)}.}
Thus,
d
d
s
∑
m
|
m
|
s
=
−
∑
n
log
(
q
n
)
q
n
(
1
−
s
)
=
−
∑
n
∑
deg
(
f
)
=
n
log
(
q
n
)
q
−
n
s
=
−
∑
f
log
|
f
|
|
f
|
−
s
.
{\displaystyle {\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}.}
We got that:
D
Λ
A
(
s
)
=
−
ζ
A
′
(
s
)
ζ
A
(
s
)
{\displaystyle D_{\Lambda _{A}}(s)={\frac {-\zeta '_{A}(s)}{\zeta _{A}(s)}}}
Now,
∑
m
Λ
A
(
m
)
|
m
|
−
s
=
∑
n
(
∑
deg
(
m
)
=
n
Λ
A
(
m
)
q
−
s
m
)
=
∑
n
(
∑
deg
(
m
)
=
n
Λ
A
(
m
)
)
u
n
=
−
ζ
A
′
(
s
)
ζ
A
(
s
)
=
q
1
−
s
log
(
q
)
1
−
q
1
−
s
=
log
(
q
)
∑
n
=
1
∞
q
n
u
n
{\displaystyle {\begin{aligned}\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=1}^{\infty }q^{n}u^{n}\end{aligned}}}
Hence,
∑
deg
(
m
)
=
n
Λ
A
(
m
)
=
q
n
log
(
q
)
,
{\displaystyle \sum _{\deg(m)=n}\Lambda _{A}(m)=q^{n}\log(q),}
q
n
{\displaystyle q^{n}}
Ave
n
Λ
A
(
m
)
=
log
(
q
)
=
1.
{\displaystyle {\text{Ave}}_{n}\Lambda _{A}(m)=\log(q)=1.}
Polynomial Euler totient function
edit
Define Euler totient function polynomial analogue,
Φ
{\displaystyle \Phi }
(
A
/
f
A
)
∗
{\displaystyle (A/fA)^{*}}
∑
deg
f
=
n
,
f
monic
Φ
(
f
)
=
q
2
n
(
1
−
q
−
1
)
.
{\displaystyle \sum _{\deg f=n,f{\text{ monic}}}\Phi (f)=q^{2n}(1-q^{-1}).}
See also
edit
References
edit
Hardy, G. H. ; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers . Revised by D. R. Heath-Brown and J. H. Silverman . Foreword by Andrew Wiles . (6th ed.). Oxford: Oxford University Press . ISBN 978-0-19-921986-5 . MR 2445243. Zbl 1159.11001.Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory . Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press . pp. 36–55. ISBN 0-521-41261-7 . Zbl 0831.11001. Tom M. Apostol (1976), Introduction to Analytic Number Theory Undergraduate Texts in Mathematics , ISBN 0-387-90163-9 Michael Rosen (2000), Number Theory in Function Fields , Springer Graduate Texts In Mathematics, ISBN 0-387-95335-3 Hugh L. Montgomery; Robert C. Vaughan (2006), Multiplicative Number Theory , Cambridge University Press, ISBN 978-0521849036 Michael Baakea; Robert V. Moodyb; Peter A.B. Pleasantsc (2000), Diffraction from visible lattice points and kth power free integers , Discrete Mathematics- Journal 
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