# Notes on probability measures on metric spaces

### 4 The Prohkorov metic

Let $(X,d)$ be a metric space. Denoted by $P(X)$ all the Borel probability measure on $X$.

We have defined the notation  of weak convergence in $P(X)$, Define for $\mu,\nu\in P(X)$

$\displaystyle d_P(\mu,\nu):=\inf\{\alpha>0:\mu(A)\leq \nu(A_\alpha)+\alpha\ \text{and}\ \nu(A)\leq \mu(A_\alpha)+\alpha\}$

where

$A_\alpha=\{x:d(x,A)<\alpha\}.$

The function $d_P$ is called the Prokhorov metric on $P(X)$ (induced by  $d$).

Conclusion If $(X,d)$ is a separable metric space, then so is $P(X)$ with the induced Prokhorov metric. Moreover,  a sequence in $P(X)$ converges in metric if and only if it converges weakly and to the same limit.

# Inequality for 1/zeta(s), zeta'(s)/zeta(s), and log zeta(s)

Theorem We have

$\displaystyle \zeta(s)=O(\log t)$

uniformly in the region

$\displaystyle 1-\frac{A}{\log t}\leq \sigma\leq 2,\ t>t_0$

where $A$ is any positive constant. In particular

$\displaystyle \zeta(1+it)=O(\log t).$

It is also easy to see that

$\displaystyle \zeta'(s)=O(\log^2 t)$

in the above region.

There is an alternative method, due to Landau obtaining results of this kind.

Lemma If $f(s)$ is analytic, and

$\displaystyle |f(s)|\leq |f(s_0)|e^M\quad (M>1)$

in the circle $|s-s_0|\leq r$, then

$\displaystyle |\frac{f'(s)}{f(s)}-\sum_{\rho}\frac{1}{s-\rho}|\leq \frac{AM}{r}\quad (|s-s_0|\leq \frac{1}{4}r),$

where $\rho$ runs through the zeros of $f(s)$ such that $|\rho-s_0|\leq \frac{1}{2}r$.

Lemma 1  If $f(s)$ satisfies the conditions of the previous lemma, and has no zeros in the right hand half of the circle $|s-s_0|\leq r$, then

$\displaystyle -\mathrm{Re}(\frac{f'(s_0)}{f(s_0)})\leq \frac{AM}{r};$

while if $f(s)$ has a zero $\rho_0$ between $s_0-\frac{1}{r}$ and $s_0$, then

$\displaystyle -\mathrm{Re}(\frac{f'(s_0)}{f(s_0)})\leq \frac{AM}{r}-\frac{1}{s_0-\rho_0}.$

Lemma Let $f(s)$ satisfy the conditions of Lemma 1, and let

$\displaystyle |\frac{f'(s_0}{f(s_0)}|\leq \frac{M}{r}.$

Suppose also that $f(s)\neq 0$ in the part $\sigma \geq\sigma_0-2r'$ of the circle $|s-s_0|\leq r$, where $0. Then

$\displaystyle |\frac{f'(s)}{f(s)}|\leq A\frac{M}{r}\quad (|s-s_0|\leq r').$

# Jensen’s formula

Jensen’s formula relates the average magnitude of an analytic function on a circle with number of its zeroes inside the circle.

Theorem (Jensen’s formula) Suppose that $f$ be an analytic function in a region in the complex plane which contains the closed disk $D$ of radius $r$ about the origin, $a_1,a_2,\dots,a_n$ are zeros of $f$ in the interior of $D$ according to multiplicity, and $f(0)\neq 0$. Then

$\displaystyle \log |f(0)|=\sum_{k=1}^n \log \frac{|a_k|}{r}+\frac{1}{2\pi}\int_0^{2\pi} \log f(re^{i\theta})\,d\theta.$

# Primitive Dirichlet Characters

Let $\chi$ be a Dirichlet characters of modulus $q$. Thus $\chi$ is a periodic function of period $q$. It is possible, that for values of $a$ restricted by condition $\gcd(a,q)=1$, the function $\chi$ may have d period less than $q$.

Definition (Primitive characthers) We say that a Dirichlet character $\chi$  of modulus $q$ is imprimitive if $\chi(a)$ restricted by $\gcd(a,q)=1$ has a period which is less than $q$, Otherwise we say that $\chi$ is primitive.

For maximal clarity, let us write out exactly what we mean $\chi(a)$ restricted $(a,q)=1$ has a period $n\in\mathbf{Z}^+$. This means: For any integers $a,b$ with $\gcd(a,q)=\gcd(b,q)=1$ and $a=b\ \text{mod}\ n$ we have $\chi(a)=\chi(b)$.

Definition (Conductor) Given a Dirichlet character $\chi$ of modulus $q$, the conductor of $\chi$ is defined to be the smallest positive integer $q'$ such that $\chi(a)$ restricted by $\gcd(a,q)=1$ has a period $q'$.

Lemma If $\chi$ is a Dirichlet character of modulus $q$ and if $q_1\in\mathbf{Z}^+$ is a period of $\chi(a)$ restricted by $\gcd(a,q)=1$, then $c(\chi)\mid q_1$. Hence in particular ,we have $c(\chi)\mid q$.

Lemma For each Dirichlet character $\chi$ of modulus $q$, there is a unique Dirichlet character of $\chi'$ of modulus $c(\chi)$ such that

$\displaystyle \chi(n)=\begin{cases}\chi'(n) & \text{if}\ \gcd(a,q)=1\\ 0 & \text{if} \gcd(a,q)>1. \end{cases}$

This Dirichlet character $\chi'$ is primitive

# 做数学一定要是天才吗？（陶哲轩）

[转载自刘小川WordPress]

，甚至更有创造性。但是，并不是这些所谓的“最好”的数学家才应该做数学。这其实是一种关于绝对优势和相对优势的很普遍的错误观念。有意义的数学科研的领

# Convergence of random variable

In this post, we will pursue a approach to the strong law of large numbers based on the convergence of random variables.

Definition (Tail $\sigma$-algebra) Let  $(X_n)_{n\geq 1}$ be a sequence of random variables. Let $\mathcal{F}_n=\sigma(X_n,X_{n+1},\dots)$ be the smallest $\sigma$-algebra with respect to which all $X_n:m\geq n$ are measure. The tail $\sigma$-algebra $\mathcal{T}:=\cap_n \mathcal{F}_n$ is the intersection of all $\mathcal{F}_n$.

Theorem (Kolmogorov’s 0-1 law) If $X_1,X_2,\dots$ are independent and $A\in\mathcal{T}$, then $P(A)=0$ or $1$.

Theorem (Kolmogorov’s maximal inequality) Suppose that $X_1,\dots,X_n$ are independent with $\mathbf{E}X_i=0$ and $\mathbf{Var}X_i<\infty$. If $S_n=X_1+\cdots+X_n$ then

$\displaystyle \mathbf{P}\left(\max_{1\leq k\leq n}|S_k|\geq x\right)\leq \frac{\mathbf{Var}S_n}{x^2}.$

Theorem Suppose that $X_1,X_2,X_3,\dots$ are independent and have $\mathbf{E}X_n=0$. If

$\displaystyle \sum_{n=1}^\infty \mathbf{Var} X_n<\infty,$

then with probability one $\sum_{n=1}^\infty X(\omega)$ converges.