The charm of Eulerian mathematics lies in its use of analogy to connect seemingly unrelated fields, leading to correct results in a vivid and ingenious manner. His calculation of \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...=\frac{\pi^2}{6} is a classic example. Although the demonstration process may not always be rigorous, those “magical” derivations are already breathtaking and often thought-provoking. This effect is usually difficult to achieve through strict formal proof; it not only provides us with answers but also gives us inspiration: new ideas, new directions; sometimes, it even reveals the inherent and profound connections between various disciplines. Below, let us observe the “Riemann \zeta function” and the “Golden Key” in number theory!
The Riemann \zeta function is defined as: \zeta (s)=\sum_{n=1}^{\infty} \frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+... Originally, s was intended to be a real number, but after introducing complex analysis into number theory, extending s to the complex plane has proven to be of even greater research value.
Regarding the Riemann \zeta function, there is a famous problem known as the “Riemann Hypothesis”: it conjectures that the real part of all non-trivial zeros of the Riemann \zeta function is 1/2. This article only mentions this problem without further extension; interested readers may refer to Wikipedia - Riemann Zeta Function and Wikipedia - Riemann Hypothesis. Although Riemann’s \zeta function is considered by mathematicians to be primarily related to the “purest” field of mathematics, number theory, it also appears in applied statistics, the mathematical theory of musical tuning, and even relates to quantum chaos in theoretical physics!
Here we focus only on the formal properties of the \zeta function. Do readers remember the most primitive method for generating prime numbers—the Sieve of Eratosthenes? Among the first n natural numbers, cross out all multiples of primes not exceeding \sqrt{n} (except for the primes themselves), and the remaining numbers are all primes. We will use a method similar to the Sieve of Eratosthenes to process the Riemann \zeta function.
\zeta (s)=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+\frac{1}{6^s}...
\frac{1}{2^s} \zeta (s)=\frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}+\frac{1}{8^s}+\frac{1}{10^s}+...
Subtracting the two: (1-\frac{1}{2^s})\zeta (s)=1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{9^s}...
In other words, all terms with even denominators have been removed. Does this not resemble the first step of the Sieve of Eratosthenes: removing all multiples of 2? Of course, the difference is that in the Sieve of Eratosthenes, 2 itself is not removed, whereas here it is. Next, we perform a similar treatment:
\frac{1}{3^s}(1-\frac{1}{2^s})\zeta (s)=\frac{1}{3^s}+\frac{1}{9^s}+\frac{1}{15^s}+\frac{1}{21^s}+\frac{1}{27^s}...
Then subtracting again yields: (1-\frac{1}{3^s})(1-\frac{1}{2^s})\zeta (s)=1+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\frac{1}{13^s}...
Repeating this process up to some sufficiently large prime number (such as 997), we obtain: \begin{aligned}(1-\frac{1}{997^s})...(1-\frac{1}{5^s})(1-\frac{1}{3^s})(1-\frac{1}{2^s})\zeta (s) \\ =1+\frac{1}{1009^s}+\frac{1}{1013^s}+\frac{1}{1019^s}+\frac{1}{1021^s}...\end{aligned}
If s > 1, then the right side converges, and as the prime numbers increase, the right side becomes smaller and smaller. Thus, we have reason to believe in the existence of the following expression: ...(1-\frac{1}{p^s})...(1-\frac{1}{5^s})(1-\frac{1}{3^s})(1-\frac{1}{2^s})\zeta (s)=1
Where the left side ranges over all prime numbers. We rewrite it as: \zeta (s)=(1-2^{-s})^{-1}(1-3^{-s})^{-1}(1-5^{-s})^{-1}(1-7^{-s})^{-1}(1-11^{-s})^{-1}...
Or, using more professional notation: \zeta (s)=\prod_{p} (1-p^{-s})^{-1}
It reveals a relationship between natural numbers and prime numbers, which is known in number theory as the “Golden Key” (actually called the Euler Product Formula, first derived by Euler). Because of it, many doors in mathematics can be opened, especially establishing connections between various mathematical fields. In fact, we have already used part of it here. Using it, we can calculate many difficult problems, such as the divergence of the sum of the reciprocals of all primes, the probability that any two natural numbers are coprime, and so on...
Reference: Prime Obsession
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