数学代写|Math46400 Number theory

Recall that for a number field $K$, the Dedekind zeta function $\zeta_K(s)$ is defined by
$$
\zeta_K(s):=\sum_I \mathrm{~N}(I)^{-s}
$$
where $I$ ranges over nonzero ideals of $\mathcal{O}_K$ and $\mathrm{N}(I)$ is the absolute norm, which is just the cardinality of the residue field $\mathcal{O}_K / \mathfrak{p}$ when $I$ is a prime ideal $\mathfrak{p}$.

The definition of $\mathrm{N}(\mathfrak{p}):=# \mathcal{O}_K / \mathfrak{p}$ as the cardinality of the residue field makes sense in any global field and extends multiplicatively to all $\mathcal{O}_K$-ideals. In this problem you will investigate the zeta function $\zeta_q(s):=\sum_I \mathrm{~N}(I)^{-s}$, where $I$ ranges over ideals of the ring of integers $\mathcal{O}_K:=\mathbb{F}_q[t]$ of the rational function field $K=\mathbb{F}_q(t)$.

For integers $n \geq 0$, the Bernoulli polynomials $B_n(x) \in \mathbb{Q}[x]$ are defined as the coefficients of the exponential generating function
$$
E(t, x):=\frac{t e^{t x}}{e^t-1}=\sum_{n \geq 0} \frac{B_n(x)}{n !} t^n .
$$
The Bernoulli numbers $B_n \in \mathbb{Q}$ are defined by $B_n=B_n(0)$.
(a) Prove that $B_0(x)=1, B_n^{\prime}(x)=n B_{n-1}(x)$, and $B_n(1)=B_n(0)$ for $n \neq 1$, and that these properties uniquely determine the Bernoulli polynomials.
(b) Prove that $B_n(x+1)-B_n(x)=n x^{n-1}$ and
$$
B_n(x+y)=\sum_{k=0}^n\left(\begin{array}{l}
n \
k
\end{array}\right) B_k(x) y^{n-k} .
$$
Use this to show that $B_k$ can alternatively be defined by the recurrence $B_0=1$ and
$$
B_n=-\frac{1}{n+1} \sum_{k=0}^{n-1}\left(\begin{array}{c}
n+1 \
k
\end{array}\right) B_k
$$
for all $n>0$, and show that $B_n=0$ for all odd $n>1$.
(c) Recall the hyperbolic cotangent function $\operatorname{coth} z:=\frac{e^z+e^{-z}}{e^z-e^{-z}}$. Prove that
$$
z \operatorname{coth} z=\sum_{n \geq 0} B_{2 n} \frac{(2 z)^{2 n}}{(2 n) !} .
$$
(d) Show that $\cot z=i \operatorname{coth} i z$ and then derive (as Euler did) the identity
$$
z \cot z=1-2 \sum_{k>1} \frac{z^2}{k^2 \pi^2-z^2}
$$
(e) Use (c) and (d) to prove that for all $n \geq 1$ we have
$$
\zeta(2 n)=(-1)^{n-1} \frac{(2 \pi)^{2 n} B_{2 n}}{2 \cdot(2 n) !}
$$
and then use the functional equation to prove that for all $n \geq 1$ we have
$$
\zeta(-n)=-\frac{B_{n+1}}{n+1}
$$
(f) Prove (rigorously!) that for any integer $n>1$ the asymptotic density of integers that are $n$-power free (not divisible by $p^n$ for any prime $p$ ) is $1 / \zeta(n)$ and compute this density explicitly for $n=2,4,6$.
(g) Prove that for all integer $n, N>1$ we have
$$
\sum_{m=0}^{N-1}(m+x)^{n-1}=\frac{B_n(N+x)-B_n(x)}{n} .
$$
Use this to deduce Faulhaber’s formula
$$
P_n(N):=\sum_{m=1}^{N-1} m^n=\frac{1}{n+1} \sum_{k=0}^n\left(\begin{array}{c}
n+1 \
k
\end{array}\right) B_k N^{n+1-k}
$$
for summing $n$th powers. Compute the polynomials $P_n(N)$ explicitly for $n=2,3,4$.

Recall that an arithmetic function is a function $a: \mathbb{Z}_{\geq 1} \rightarrow \mathbb{C}$; we say that $a \neq 0$ is multiplicative if $a(m n)=a(m) a(n)$ holds for all relatively prime $m, n$, and totally multiplicative if this holds for all $m, n$. Below are some examples; as usual, $p$ denotes a prime, $p^e$ denotes a (nontrivial) prime power, and $d \mid n$ indicates that $d$ is a positive divisor of $n$.

$0(n)=0,1(n)=1, \operatorname{id}(n):=n, \quad e(n):=0^{n-1}$;

$\tau(n):=#{d \mid n}, \quad \sigma(n):=\sum_{d \mid n} d$

$\omega(n):=#{p \mid n}, \Omega(n):=#\left{p^e \mid n\right}, \phi(n):=#(\mathbb{Z} / n \mathbb{Z})^{\times}$;

$\lambda(n):=(-1)^{\Omega(n)}, \mu(n):=(-1)^{\omega(n)} \cdot 0^{\Omega(n)-\omega(n)}, \quad \mu^2(n):=\mu(n)^2$.
The set of all arithmetic functions forms a $\mathbb{C}$-vector space that we denote $\mathcal{A}$. Associated to each arithmetic function is a Dirichlet series $\sum_{n \geq 1} c_n n^{-s}$ defined by
$$
D_a(s):=\sum_{n \geq 1} a(n) n^{-s}
$$
The Dirichlet convolution $a * b$ of arithmetic functions $a$ and $b$ is defined by
$$
(a * b)(n):=\sum_{d \mid n} a(d) b(n / d)
$$
We use $f^{* n}$ to denote the $n$-fold convolution $f * \cdots * f$.

(a) Prove that for any arithmetic functions we have $D_{a * b}=D_a D_b$ and that endowing $\mathcal{A}$ with a multiplication defined by Dirichlet convolution makes $\mathcal{A}$ a $\mathbb{C}$-algebra that is isomorphic to the $\mathbb{C}$-algebra of Dirichlet series (with the usual multiplication).
(b) Show that $\mathcal{A}$ is a local ring with unit group $\mathcal{A}^{\times}={f \in \mathcal{A}: f(1) \neq 0}$ and maximal ideal $\mathcal{A}0={f \in \mathcal{A}: f(1)=0}$. Prove that the set of multiplicative functions $\mathcal{M}$ forms a subgroup of $\mathcal{A}_1:={f \in \mathcal{A}: f(1)=1} \subseteq \mathcal{A}^{\times}$. Is this also true of the set of totally multiplicative functions? (c) Prove the following identities $\mu * 1=e, \phi * 1=\mathrm{id}, \mu * \mathrm{id}=\phi, 1 * 1=\tau, \mathrm{id} * 1=\sigma$. Use $\mu * 1=e$ to give a one-line proof of the Möbius inversion formula. (d) Define the exponential map $\exp : \mathcal{A} \rightarrow \mathcal{A}$ by $$ \exp (f):=\sum{n=0}^{\infty} \frac{f^{* n}}{n !}=e+f+\frac{f * f}{2}+\cdots
$$
Prove that exp defines a group isomorphism from $\left(\mathcal{A}0,+\right)$ to $\left(\mathcal{A}_1, \right)$ with inverse $$ \log (f):=\sum{n=1}^{\infty} \frac{(-1)^{n-1}(f-e)^{ n}}{n}
$$
(e) Define $\kappa(n)$ to be $1 / k$ if $n=p^k$ is a prime power and 0 otherwise. Prove that $\exp \kappa=1$, and deduce that $\exp (-\kappa)=\mu$ and $\exp (2 \kappa)=\tau$.
(f) Prove that each $f \in \mathcal{A}_1$ has a unique square-root $g \in \mathcal{A}_1$ for which $g^{* 2}=f$ that we denote $f^{* 1 / 2}$. Prove that $1^{* 1 / 2}=\exp (\kappa / 2)$ and compute $\exp (\kappa / 2)(n)$ for $n$ up to 10 .