# 【校理】为了人类心智的荣耀

## 一.

### 1.1

$$r(n,k)=\sum_{s+t=n\\s,t>0}r(s,j)r(t,j)~\mbox{for some}~i+j=k.$$

$$\theta(\tau,k)=\sum_{n=0}^\infty r(n,k)q^n~\mbox{where}~q=e^{2\pi i\tau},\tau\in\mathcal{H},$$

$$\theta(\tau,k_1)\theta(\tau,k_2)=\theta(\tau,k_1+k_2).$$

$$\theta(\tau+1,k)=\theta(\tau,k).$$

$$\theta(\frac{\tau}{4\tau+1},1)=\sqrt{4\tau+1}\theta(\tau,1).$$

$$\theta(\frac{\tau}{4\tau+1},k)=(4\tau+1)^{k/2}\theta(\tau,k).$$

### 1.2

$$f(\gamma(\tau))=(c\tau+d)^kf(\tau)\mbox{for any}\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma\subset SL_2(\mathbb{Z}).$$

$$\pm\begin{pmatrix}1&1\\0&1\end{pmatrix},\pm\begin{pmatrix}1&0\\4&1\end{pmatrix}.$$

$$\Gamma_0(4)=\{ \begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL_2(\mathbb{Z}):\begin{pmatrix}a&b\\c&d\end{pmatrix}\equiv\begin{pmatrix}\ast&\ast\\0&\ast\end{pmatrix}(\mathrm{mod~}4)\}.$$

$$\Gamma_1(4)=\{ \begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL_2(\mathbb{Z}):\begin{pmatrix}a&b\\c&d\end{pmatrix}\equiv\begin{pmatrix}1&\ast\\0&1\end{pmatrix}(\mathrm{mod}~4)\}.$$

$$\theta(\tau,k)\in\mathcal{M}_k(\Gamma_0(4))\subset\mathcal{M}_k(\Gamma_1(4)).$$

### 1.3

$$D=\sum_{x\in X}n_xx,n_x\in\mathbb{Z},n_x=0~\mbox{for all but finite}x.$$

$$\mathrm{div}(f)=\sum_{x\in X}v_x(f)x,$$

$$L(D)=\{f\in C(X):f=0~\mbox{or}~\mathrm{div}(f)+D\geq0,\}$$

\begin{align*}\mathcal{A}_k(\Gamma)&=\{f_0f\in\mathcal{A}_k(\Gamma):f_0f=0\mbox{or}\mathrm{div}(f_0f)\geq0\}\\&\cong\{f_0\in C(x(\Gamma)):f_0=0~\mbox{or}~\mathrm{div}(f_0)+\mathrm{div}(f)\geq0\}\\&=\{f_0\in C(X(\Gamma)):f_0=0~\mbox{or}~\mathrm{div}(f_0)+\left \lfloor\mathrm{div}(f)\right\rfloor\geq0\}\\&=L(\left\lfloor\mathrm{div}(f)\right\rfloor)\end{align*}

$$\mathcal{M}_k(\Gamma)\cong L(\left\lfloor\mathrm{div}(f)\right\rfloor.$$

### 1.4

$$l(D)=\mathrm{deg}(D)-g+1.$$

\begin{align*}\mathrm{dim}(\mathcal{M}_1(\Gamma_1(4)))&=1\\ \mathrm{dim}(\mathcal{M}_2(\Gamma_1(4)))&=2\\ \mathrm{dim}(\mathcal{M}_3(\Gamma_1(4)))&=2\\ \mathrm{dim}(\mathcal{M}_4(\Gamma_1(4)))&=3\end{align*}

### 1.5

$$\theta(\tau,2)\in\mathcal{M}_1(\Gamma_1(4)).$$

（就以它为例吧，因为这个空间是1维的）所以只要找到右边这个1维空间的一个基，特别的，就是其中的一个非零函数，$\theta(\tau, 2)$与它最多相差一个复系数！

Eisenstein series:

$$G_k(\tau)=\sum_{(c,d)\in\mathbb{Z}^2}\frac{1}{(c\tau+d)^k},\tau\in\mathcal{H}.$$

Riemann zeta function:

$$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},~\mathrm{Re}(s)>1.$$

Hurwitz zeta function:

$$\zeta_{+}^n(s)=\sum_{m=1\\m\equiv n(N)}\frac{1}{m^s},~n\in\mathbb{Z}_N^\ast.$$

Mobius function:

$$\mu(n)=\begin{cases}0,~&\mbox{if}~p^2|n~\mbox{for some prime}~p\\(-1)^g,~&\mbox{if}~n=p_1\cdots p_g~\mbox{for distinct primes}\end{cases}$$

Dirichlet character: 一个乘法群的同态

$$\chi=:\mathbb{Z}_N^\ast\rightarrow\mathbb{C}$$

Gamma function:

$$\Gamma(s)=\int_0^\infty e^{-t}t^{s-1}\mathrm{d}t,~s\in\mathbb{C},\mathrm{Re}(s)>0.$$

Dirichlet L-function:

$$L(s,\chi)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s},~\mathrm{Re}(s)>1.$$

Weierstrass sigma-function:

$$\sigma_\Lambda(z)=z\prod_{0\neq\omega\in\Lambda}(1-\frac{z}{\omega})e^{\frac{z}{\omega}+\frac{1}{2}(\frac{z}{\omega})^2}.$$

……实在是列不动了。。。

$$E_1^{\psi,\varphi}(\tau)=\delta(\varphi)L(0,\psi)+\delta(\psi)L(0,\varphi)+2\sum_{n=1}^\infty\sigma_0^{\psi,\varphi}(n)q^n.$$

$$\sigma_{k-1}^{\psi,\varphi}=\sum_{0<m|n}\psi(\frac{n}{m})\varphi(m)m^{k-1}.$$

$$r(n,2)=C\sigma_0^{\chi,\mathbf{1}}(n),$$

$$r(n,2)=4\sum_{0<m|n\\m~\mbox{odd}}(-1)^{\frac{m-1}{2}},$$

$$r(n,4)=8\sum_{0<d|n\\4\nmid d}d.$$

## 二.

1）如果我带你爬山之前，先用直升飞机带你慢慢的飞到山顶，让你先欣赏一下沿途的风景以及从山顶往山下俯瞰的美景，再带你从头开始爬。

2）我告诉你我们去爬一个不知名的山吧！然后从山脚开始费力的攀登。

## 三.

1）数学家们的工作似乎没做错。不然两个分支撞在一起矛盾了就麻烦了。

2）整个数学并没有还原出那个与生俱来的intrinsic的结构的本来面貌。肯定还有一个更基本更底层的结构不为人所知。

## 四.

posted @ 2018-04-14 23:21  黑山雁  阅读(888)  评论(0编辑  收藏  举报