Prime Dream（3）——Mertens的几个渐进公式

2022-03-05 08:20 作者:子瞻Louis 0人读过 | 我要投稿

上一节里扯到了下面这样的渐进公式：

#） $%5Csum_%7Bp%5Cle%20x%7D%5Cfrac%7B%5Clog%20p%7D%7Bp%7D%3D%5Clog%20x%2B%5Cmathcal%20O(1)$

它被称为Mertens第一定理，通过右式发散便能得到Euclid定理，此外，可以由他出发得到另外的与素数有关的渐进公式。本文出现的两个公式是由Mertens本人提出的，它们与Mertens第一定理可以被共同称为Mertens公式，而最后一个公式又被称为Mertens第二定理

素数的倒数和

引入一个常数——Mertens常数

$M%3D%5Clim_%7Bx%5Cto%5Cinfty%7D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p-%5Clog%5Clog%20x$

它在定义上与Euler常数十分类似，不过我们需要说明这个式子是有意义的，即要说明右侧确实是收敛到某个常数的。为了简便，这里会采用以下记号：

$%5Cint_%7Bc-%7D%5Ex%3A%3D%5Clim_%7B%5Cdelta%5Cto%2B0%7D%5Cint_%7Bc-%5Cdelta%7D%5Ex$

首先令

$R(x)%3A%3D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac%7B%5Clog%20p%7Dp-%5Clog%20x$

由Mertens第一定理可知 $R(x)%3D%5Cmathcal%20O(1)$ .根据Riemann-Stieltjes积分的分部积分，有

$%5Cbegin%7Baligned%7D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%26%3D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1%7B%5Clog%20p%7D%5Ccdot%5Cfrac%7B%5Clog%20p%7Dp%5C%5C%26%3D%5Cint_%7B2-%7D%5Ex%5Cfrac1%7B%5Clog%20t%7D%5Cmathrm%20d%5Csum_%7Bp%5Cle%20t%7D%5Cfrac%7B%5Clog%20p%7Dp%5C%5C%26%3D%5Cint_%7B2-%7D%5Ex%5Cfrac1%7B%5Clog%20t%7D%5Cmathrm%20d%5Clog%20t%2B%5Cint_%7B2-%7D%5Ex%5Cfrac1%7B%5Clog%20t%7D%5Cmathrm%20dR(t)%5C%5C%26%3D%5Clog%5Clog%20x-%5Clog%5Clog2%2B%5Cfrac%7BR(x)%7D%7B%5Clog%20x%7D-%5Cfrac%7BR(2)%7D%7B%5Clog2%7D%2B%5Cint_%7B2-%7D%5Ex%5Cfrac%7BR(t)%7D%7Bt%5Clog%5E2t%7D%5Cmathrm%20dt%5Cend%7Baligned%7D$

于是，可得

$%5Clim_%7Bx%5Cto%5Cinfty%7D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p-%5Clog%5Clog%20x%3D1-%5Clog%5Clog2%2B%5Cint_%7B2-%7D%5E%5Cinfty%5Cfrac%7BR(t)%7D%7Bt%5Clog%5E2t%7D%5Cmathrm%20dt$

显然右式收敛到一个常数，即Mertens常数，将它代回式中，可得：

$%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%3D%5Clog%5Clog%20x%2BM%2B%5Cfrac%7BR(x)%7D%7B%5Clog%20x%7D-%5Cint_%7Bx%7D%5E%5Cinfty%5Cfrac%7BR(t)%7D%7Bt%5Clog%5E2t%7D%5Cmathrm%20dt$

其中

$%5Cleft%7C%5Cint_%7Bx%7D%5E%5Cinfty%5Cfrac%7BR(t)%7D%7Bt%5Clog%5E2t%7D%5Cmathrm%20dt%5Cright%7C%5Cle%20%5Cleft%7C%5Cint_%7Bx%7D%5E%5Cinfty%5Cfrac1%7Bt%5Clog%5E2t%7D%5Cmathrm%20dt%5Cright%7C%3D%5Cfrac1%7B%5Clog%20x%7D$

因此，我们得到素数倒数和的渐进公式：

1） $%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%3D%5Clog%5Clog%20x%2BM%2B%5Cmathcal%20O%5Cleft(%5Cfrac1%7B%5Clog%20x%7D%5Cright)$

在上式中令 $x%3De%5Et$ ，则有

$%5Csum_%7Bp%5Cle%20e%5Et%7D%5Cfrac1p%3D%5Clog%20t%2BM%2B%5Cmathcal%20O%5Cleft(%5Cfrac1t%5Cright)$

对比自然数的倒数和

$%5Csum_%7Bn%5Cle%20t%7D%5Cfrac1n%3D%5Clog%20t%2B%5Cgamma%2B%5Cmathcal%20O%5Cleft(%5Cfrac1t%5Cright)$

发现他们异常相似。没错，数学就是这么神奇！

Mertens公式

考虑zeta函数的欧拉乘积的对数，分离出素zeta函数

$%5Cbegin%7Baligned%7D%5Clog%5Czeta(s)%26%3D%5Csum_%7Bp%7D%5Clog%5Cfrac1%7B1-p%5E%7B-s%7D%7D%5C%5C%26%3D%5Csum_%7Bp%7D%5Cfrac1%7Bp%5E%7Bs%7D%7D%2B%5Csum_%7Bp%7D%5Cleft(%5Clog%5Cfrac1%7B1-p%5E%7B-s%7D%7D-%5Cfrac1%7Bp%5Es%7D%5Cright)%5Cend%7Baligned%7D$

那么分离出来它有什么用呢？我们都知道这个等式在s=1处是发散的，但是，第二个和式在s=1时是收敛的：

$%5Cbegin%7Baligned%7D%5Csum_%7Bp%3Ex%7D%5Cleft(%5Clog%5Cfrac1%7B1-p%5E%7B-1%7D%7D-%5Cfrac1%7Bp%7D%5Cright)%26%3D%5Csum_%7Bp%3Ex%7D%5Csum_%7Bn%3D2%7D%5E%5Cinfty%5Cfrac1%7Bnp%5E%7Bn%7D%7D%5C%5C%26%5Cle%5Cfrac12%5Csum_%7Bp%3Ex%7D%5Csum_%7Bn%3D2%7D%5E%5Cinfty%5Cfrac1%7Bp%5E%7Bn%7D%7D%5C%5C%26%3D%5Cfrac12%5Csum_%7Bp%3Ex%7D%5Cleft(%5Cfrac1%7Bp-1%7D-%5Cfrac1p%5Cright)%5C%5C%26%3C%5Csum_%7Bn%3Ex%20%7D%5Cleft(%5Cfrac1%7Bn-1%7D-%5Cfrac1n%5Cright)%3D%5Cfrac1%7B%5Bx%5D%7D%5Cend%7Baligned%7D$

接着研究左侧式子，令 $s%3D1%2B%5Cepsilon%2C(0%3C%5Cepsilon%3C1)$ ，有

$%5Cbegin%7Baligned%7D%5Czeta(1%2B%5Cepsilon)-%5Cfrac1%5Cepsilon%26%3D%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac1%7Bn%5E%7B1%2B%5Cepsilon%7D%7D-%5Cint_1%5E%5Cinfty%5Cfrac1%7Bu%5E%7B1%2B%5Cepsilon%7D%7D%5Cmathrm%20du%5C%5C%26%3D%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cint_n%5E%7Bn%2B1%7D%5Cfrac1%7Bn%5E%7B1%2B%5Cepsilon%7D%7D-%5Cfrac1%7Bu%5E%7B1%2B%5Cepsilon%7D%7D%5Cmathrm%20du%5C%5C%26%5Cle%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac1%7Bn%5E%7B1%2B%5Cepsilon%7D%7D-%5Cfrac1%7B(n%2B1)%5E%7B1%2B%5Cepsilon%7D%7D%5Cle1%5Cend%7Baligned%7D$

由此得到

$%5Clog%5Czeta(1%2B%5Cepsilon)%3D%5Clog%5Cleft(%5Cfrac1%5Cepsilon%2B%5Cmathcal%20O(1)%5Cright)%3D%5Clog%5Cfrac1%5Cepsilon%2B%5Clog(1%2B%5Cmathcal%20O(%5Cepsilon))%3D%5Clog%5Cfrac1%5Cepsilon%2BO(%5Cepsilon)$

又有：

$%5Clog%5Cfrac%7B1-e%5E%7B-%5Cepsilon%7D%7D%7B%5Cepsilon%7D%3D%5Clog(1%2B%5Cmathcal%20O(%5Cepsilon))%3D%5Cmathcal%20O(%5Cepsilon)$

于是

$%5Cbegin%7Baligned%7D%5Clog%5Czeta(1%2B%5Cepsilon)%26%3D%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D%2B%5Clog%5Cfrac%7B1-e%5E%7B-%5Cepsilon%7D%7D%7B%5Cepsilon%7D%2B%5Cmathcal%20O(%5Cepsilon)%5C%5C%26%3D%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D%2B%5Cmathcal%20O(%5Cepsilon)%5Cend%7Baligned%7D$

代入到分离素zeta函数后的欧拉乘积中，

$%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D%2B%5Cmathcal%20O(%5Cepsilon)%3D%5Csum_%7Bp%7D%5Cfrac1%7Bp%5E%7B1%2B%5Cepsilon%7D%7D%2B%5Csum_%7Bp%7D%5Cleft(%5Clog%5Cfrac1%7B1-p%5E%7B-1-%5Cepsilon%7D%7D-%5Cfrac1%7Bp%5E%7B1%2B%5Cepsilon%7D%7D%5Cright)$

围绕上式，接下来考虑除收敛部分外的两项的差，记

$H(t)%3A%3D%5Csum_%7Bn%5Cle%20t%7D%5Cfrac1n%2CP(t)%3A%3D%5Csum_%7Bp%5Cle%20t%7D%5Cfrac1p$

由Riemann-Stieltjes积分的分部积分，

$%5Cbegin%7Baligned%7D%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D%26%3D%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Be%5E%7B-%5Cepsilon%20n%7D%7Dn%5C%5C%26%3D%5Cint_%7B0%7D%5E%5Cinfty%20e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dH(t)%5C%5C%26%3D%5Cepsilon%5Cint_%7B0%7D%5E%5Cinfty%20H(t)e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dt%5Cend%7Baligned%7D$

$%5Cbegin%7Baligned%7D%5Csum_%7Bp%7D%5Cfrac1%7Bp%5E%7B1%2B%5Cepsilon%7D%7D%26%3D%5Cint_%7B1%7D%5E%5Cinfty%5Cfrac1%7Bu%5E%5Cepsilon%7D%5Cmathrm%20dP(u)%5C%5C%26%3D%5Cint_%7B0%7D%5E%5Cinfty%20e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dP(e%5Et)%5C%5C%26%3D%5Cepsilon%5Cint_%7B0%7D%5E%5Cinfty%20P(e%5Et)e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dt%5Cend%7Baligned%7D$

由此可得

$%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D-%5Csum_%7Bp%7D%5Cfrac1%7Bp%5E%7B1%2B%5Cepsilon%7D%7D%3D%5Cepsilon%5Cint_%7B0%7D%5E%5Cinfty%20(H(t)-P(e%5Et))e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dt$

根据H(t)与P(t)的渐进公式，得

$H(t)-P(e%5Et)%3D%5Cgamma-M%2B%5Cmathcal%20O%5Cleft(%5Cfrac1t%5Cright)$

于是

$%5Cbegin%7Baligned%7D%5Clog%5Cfrac1%7B1-e%5E%7B-%5Cepsilon%7D%7D-%5Csum_%7Bp%7D%5Cfrac1%7Bp%5E%7B1%2B%5Cepsilon%7D%7D%26%3D%5Cgamma-M%2B%5Cmathcal%20O%5Cleft(%5Cepsilon%5Cint_%7B0%7D%5E%5Cinfty%20%5Cfrac1%7Bt%2B1%7D%20e%5E%7B-%5Cepsilon%20t%7D%5Cmathrm%20dt%5Cright)%5C%5C%26%3D%5Cgamma-M%2B%5Cmathcal%20O%5Cleft(%5Cepsilon%5Clog%20%5Cfrac1%5Cepsilon%5Cright)%5Cend%7Baligned%7D$

再次代回到欧拉乘积中，并令 $%5Cepsilon%5Cto%200$ ，得到

$%5Cgamma-M%3D%5Csum_%7Bp%7D%5Cleft(%5Clog%5Cfrac1%7B1-p%5E%7B-1%7D%7D-%5Cfrac1%7Bp%7D%5Cright)$

将右式分为不大于x与大于x的两部分和，由前文的

$%5Csum_%7Bp%3Ex%7D%5Cleft(%5Clog%5Cfrac1%7B1-p%5E%7B-1%7D%7D-%5Cfrac1%7Bp%7D%5Cright)%5Cll%5Cfrac1x$

以及素数倒数和的渐进公式，可得

$%5Cbegin%7Baligned%7D%5Csum_%7Bp%5Cle%20x%7D%5Clog%5Cleft(1-%5Cfrac1p%5Cright)%26%3DM_0-%5Cgamma-%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%2BO%5Cleft(%5Cfrac1%7Bx%7D%5Cright)%5C%5C%26%3D-%5Cgamma-%5Clog%5Clog%20x%2B%5Cmathcal%20O%5Cleft(%5Cfrac1%7B%5Clog%20x%7D%5Cright)%5Cend%7Baligned%7D$

取e的幂，根据

$%5Cexp%7B%5Cmathcal%20O%5Cleft(%5Cfrac1%7B%5Clog%20x%7D%5Cright)%7D%3D1%2B%5Cmathcal%20O%5Cleft(%5Cfrac1%7B%5Clog%20x%7D%5Cright)$

就能得到大名鼎鼎的Mertens公式了：

2） $%5Cprod_%7Bp%5Cle%20x%7D%5Cleft(1-%5Cfrac1p%5Cright)%3D%5Cfrac%7Be%5E%7B-%5Cgamma%7D%7D%7B%5Clog%20x%7D%5Cleft(1%2B%5Cmathcal%20O%5Cleft(%5Cfrac1%7B%5Clog%20x%7D%5Cright)%5Cright)$

Tchebyshev定理

在之前一期专栏中得到了

$A%5Cle%5Cfrac%7B%5Cpi(x)%5Clog%20x%7D%7Bx%7D%5Cle%20B$

这里就来研究一下中间函数的上确界与下确界吧，设

$l%3A%3D%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cinf%5Cfrac%7B%5Cpi(x)%5Clog%20x%7Dx%2CL%3A%3D%5Clim_%7Bx%5Cto%5Cinfty%7D%5Csup%5Cfrac%7B%5Cpi(x)%5Clog%20x%7Dx$

对 $%5Cepsilon%3E0$ ，存在 $x_0%5Cge2$ ，使得 $t%3Ex_0$ 时，

$l-%5Cepsilon%5Cle%5Cfrac%7B%5Cpi(t)%5Clog%20t%7D%7Bt%7D%5CRightarrow%5Cpi(t)%5Cge(l-%5Cepsilon)%5Cfrac%20t%7B%5Clog%20t%7D$

利用素数计数函数，有

$%5Cbegin%7Baligned%7D%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%5Cge%5Cint_%7Bx_0%7D%5Ex%5Cfrac%7B%5Cmathrm%20d%20%5Cpi(t)%7Dt%26%3D%5Cfrac%7B%5Cpi(x)%7Dx-%5Cfrac%7B%5Cpi(x_0)%7D%7Bx_0%7D%2B%5Cint_%7Bx_0%7D%5Ex%5Cfrac%7B%5Cpi(t)%7D%7Bt%5E2%7D%5Cmathrm%20dt%5C%5C%26%5Cge%20o(1)-1%2B(l-%5Cepsilon)%5Cint_%7Bx_0%7D%5Ex%5Cfrac1%7Bt%5Clog%20t%7D%5Cmathrm%20dt%5C%5C%26%3D(l-%5Cepsilon)(%5Clog%5Clog%20x-%5Clog%5Clog%20x_0)-1%2Bo(1)%5Cend%7Baligned%7D$

根据素数倒数和的渐进公式，可知 $l-%5Cepsilon%5Cle1$ 又由 $%5Cepsilon$ 任意小，得到 $l%5Cle1$ ，类似的又有

$%5Csum_%7Bp%5Cle%20x%7D%5Cfrac1p%5Cle(L%2B%5Cepsilon)(%5Clog%5Clog%20x-%0A%5Clog%5Clog%20x_0)%2B%5Cmathcal%20O(1)$

从而 $l%5Cle1%5Cle%20L$ ，即

3） $%5Clim_%7Bx%5Cto%5Cinfty%7D%5Cinf%5Cfrac%7B%5Cpi(x)%5Clog%20x%7Dx%5Cle1%5Cle%5Clim_%7Bx%5Cto%5Cinfty%7D%5Csup%5Cfrac%7B%5Cpi(x)%5Clog%20x%7Dx$

此时的结论已经够强了，如果能证明 $%5Cfrac%7B%5Cpi(x)%5Clog%20x%7D%7Bx%7D$ 的极限存在，那么其必收敛到1，然而素数定理的证明难就难在其极限存在性仅用初等方法是不容易确定的，因此人们会寻找更强更有力的方法来解决它——解析方法。不出意外的话下一期就要进入漫漫解析路了

参考

《解析与概率数论导引》by G.特伦鲍姆
Meten定理与素数定理 by TravorLZH:https://zhuanlan.zhihu.com/p/338578631

标签：

Prime Dream（3）——Mertens的几个渐进公式

素数的倒数和

Mertens公式

Tchebyshev定理

参考