Zeta Archive: 欧拉乘积公式

欧拉乘积公式

Bernoulli discovered Bernoulli numbers, which were originally used to solve the summation problem of equal powers.

伯努利发现了伯努利数，这些数最初是用于解决等幂和问题的。

The sum to equal powers problem refers to the formula for finding.

“等幂和问题” 指的是用于计算的公式。

Euler modified the above formula to become this zeta function:

伯努利的学生欧拉对上述公式进行了修改，从而得出了这个 Zeta 函数：

Let's try to apply The Sieve of Eratosthenes mentioned earlier to the zeta function.

让我们尝试将之前提到的埃拉托斯特尼筛法应用到 Zeta 函数中。

Multiply both sides of the equation by :

将方程两边同时乘以：

Subtract expression from expression to get:

用表达式减去表达式，得到：

So the right-hand side of the equation cancels out all the multiples of 2.

因此，等式右边的所有 2 的倍数都相互抵消了。

Review The Sieve of Eratosthenes.

回顾埃拉托色尼筛法。

Multiply both sides of the equation by, denoted :

将方程两边同时乘以：

All the multiples of 3 disappear. The first number on the right becomes 5. Continue to get:

所有 3 的倍数都消失了。右边的第一个数字变为 5。接着继续得到：

Do you notice that this is similar to The Sieve of Eratosthenes?

你有没有注意到这与埃拉托色尼筛法类似呢？

In fact, the difference should be noticed first.

事实上，首先应当注意到的是这种差异。

In the Sieve of Eratosthenes, leave unchanged and subtract their integer multiples in turn.

在埃拉托斯特尼筛法中，将这些数保持不变，然后依次减去它们的整数倍。

Now, we have removed the original prime ( ) and all its multiples from the right.

现在，我们已经将原始的质数（）及其所有倍数从右侧移除掉了。

If you go all the way down to a larger prime number, such as 997, you get:

如果一直向下继续找更大的质数，比如 997，那么结果就是：

If you repeat this process over and over again, you get the following results:

如果你不断重复这个过程，就会得到以下结果：

Divide both sides of the above formula repeatedly by each bracketed term in turn to get:

依次对上述公式两边的每个括号内的项进行除法运算，即可得到：

Now all that's left of this equation is something to do with prime numbers.

现在这个等式剩下的部分就与质数有关了。

To wit:

即：

The above formula is called Euler product formula.

上述公式被称为 “欧拉乘积公式”。

It's easy to see that this thing is related to prime numbers, and a bunch of products of prime numbers on the right can be converted into a bunch of summations of natural numbers on the left, which is more like a bridge to build a wonderful connection.

不难看出，这个式子与质数有关，而右边一系列的质数乘积可以转换为左边一系列的自然数求和，这就好像是搭建起一座桥梁，从而建立起一种美妙的联系。

Euler used this formula to prove that the number of prime numbers is infinite and ran away happily.

欧拉利用这个公式证明了质数的数量是无限的，然后就鸣金收兵了。

Later Riemann studied the Zeta function in detail, giving a paper "On the Number of Primes Less Than a Given Magnitude". In honor of Riemann's work, this function is called Riemann Zeta function.

后来，黎曼对 Zeta 函数进行了深入研究，并发表了一篇题为《论小于给定数值的素数个数》的论文。为了纪念黎曼的这一研究成果，这个函数被称为黎曼 Zeta 函数。