素数定理的证明大地图

以下陈列我认为素数定理的应该有的展开途径. 

1. 定义Riemann zeta函数$$zeta(s)=sum_{ngeq 1}frac{1}{n^s}qquad Re s>0 $$证明$zeta(s)-frac{1}{s-1}$是全纯的.
2. 证明如下恒等式$$Re s>1quad Rightarrow quad zeta(s)=prod_{p}frac{1}{1-frac{1}{p^s}}quadfrac{1}{zeta(s)}=sum_{ngeq 1}frac{mu(n)}{n^s}quad  -zeta'(s)=sum_{ngeq 1}frac{log n}{n^s}quad  -frac{zeta'(s)}{zeta(s)}=sum_{ngeq 1}frac{Lambda(n)}{n^s}$$其中$Lambda(n)=egin{cases}log p& n=p^k\ 0 & extrm{其他情况}end{cases}$. 
3. 证明$$sigma>1quad Rightarrow quad |zeta^3(sigma)zeta^4(sigma+it)zeta(sigma+2it)|geq 1$$从而证明$zeta(s)$在$Re s=1$这条直线上没有零点. 
4. 假设$s=sigma+it$, 任意$delta>0$, $$forall sigmageq 1,|t|geq 1, quad |zeta(s)|leq A_{delta} |t|^{delta}quad frac{1}{|zeta(s)|}leq B_{delta}|t|^{delta}quad|zeta'(s)|leq C_{delta}|t|^deltaquad  left|-frac{zeta'(s)}{zeta(s)} ight|leq D_{delta}|t|^delta$$
5. 证明$$pi(x)sim frac{x}{log x}iff sum_{nleq x} Lambda(n)sim x$$ 
6. 证明对任何$c>1$, $$displaystyle frac{1}{2pi i}int_{c-i infty}^{c+iinfty} frac{x^s}{s} extrm{d}s=egin{cases}1 & x>1\ 0 & 0<x<1end{cases}$$ 
7. 因此$$sum_{nleq x} Lambda(n)= frac{1}{2pi i}int_{c-i infty}^{c+iinfty} frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s$$
8. 经过计算$frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight)$在$s=1$的留数就是$x-1$. 
9. 选择充分大的$T$, 因为无零点, 可以选择充分接近$1$的$sigma_0$使得在$[sigma_0,1] imes [-T,T]$内$zeta(s)$没有零点, 考虑如下五条直线段$$ell_1=[1-iinfty, 1-iT]qquad ell_2=[1-iT,sigma_0-it]qquad ell_3=[sigma_0-iT,sigma_0+iT]qquad ell_4=[sigma_0+iT,1+iT]qquad ell_5=[1+iT,1+iinfty]$$因为$h(s)$增长缓慢, 利用留数定理有$$frac{1}{2pi i}int_{c-i infty}^{c+iinfty} frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s=x-1+frac{1}{2pi i}int_{ell_1+ell_2+ell_3+ell_4+ell_5}  frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s$$故只要证明$frac{1}{x}int_{ell_1+ell_2+ell_3+ell_4+ell_5}frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s o 0$. 
10. 任意$epsilon $可以取$T_epsilon$充分大, 使得$ell_{1,5}$上积分$leq epsilon x$, $ell_3$上积分$ll  x^{sigma_0}$, $ell_{2,4}$上积分$ll frac{x}{log x}$, 这样$$left|int_{ell_1+ell_2+ell_3+ell_4+ell_5}frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s ight|leq epsilon x+C_{epsilon}frac{x}{log x}$$故$$left|frac{1}{x}int_{ell_1+ell_2+ell_3+ell_4+ell_5}frac{x^s}{s}left(-frac{zeta'(s)}{zeta(s)} ight) extrm{d}s ight|leq epsilon +frac{C_{epsilon}}{log x} $$令$x$充分大, 命题得证.