In 1798, the French mathematician Legendre proposed: \pi(n) \sim \frac{n}{\ln n}
This formula is known as the "Prime Number Theorem" (PNT). What does it mean? Here, \pi(N) refers to the number of primes not greater than N, and \frac{N}{\ln N} is a calculated result. The symbol \sim means "asymptotically equivalent to." The entire expression means that "the number of primes not greater than N is asymptotically equivalent to \frac{N}{\ln N}." Simply put, it says that \frac{N}{\ln N} is an approximate estimate of \pi(N). Some readers might ask why we use \sim instead of \approx. In fact, \sim also implies: \lim_{N \to \infty} \frac{\pi(N) \ln N}{N} = 1
This is the essence of the \sim symbol; it includes the limiting case, whereas \approx does not. Like the equals sign, \sim is transitive: if A \sim B and B \sim C, then A \sim C. The Prime Number Theorem was proven in 1896 (using complex analysis), and later, several advanced or elementary proofs appeared. For details, one can refer to the Wikipedia entry on the Prime Number Theorem. The fundamental significance of the PNT lies in this: the appearance of primes seems to follow no pattern, yet viewed as a whole, the count of primes actually follows a certain law, which is truly surprising and delightful!
Corollaries of the PNT: 1. The probability that N is prime is approximately \frac{1}{\ln N}; 2. The N-th prime is approximately N \ln N. These two statements are equivalent to the Prime Number Theorem; in other words, proving one allows the derivation of the other two.
However, although the Prime Number Theorem holds, it is too coarse. For the estimation of the number of primes within 10^{12}, it produces an error of 4%. Below, we will start from the results of the previous article to derive a "strengthened version" of the Prime Number Theorem. In the previous article, we obtained: \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{p} = Q > \ln \ln (p+1) - \ln 2 and mentioned that \ln \ln p is a fairly good approximation of Q.
Making a rough estimate here, we directly write: Q_n = \ln \ln p_n Simultaneously, Q_{n+1} = Q_n + \frac{1}{p_{n+1}} = \ln \ln p_{n+1} Subtracting them gives: \frac{1}{p_{n+1}} = \ln \frac{\ln p_{n+1}}{\ln p_n} Let \Delta p = p_{n+1} - p_n. When p_n is sufficiently large, \Delta p is "insignificant." According to the approximation \ln(x + \varepsilon) \approx \ln x + \frac{\varepsilon}{x}, we can write: \begin{aligned} \frac{1}{p_{n+1}} &= \ln \frac{\ln p_{n+1}}{\ln p_n} = \ln \frac{\ln p_n + \frac{\Delta p}{p_n}}{\ln p_n} \\ &= \ln\left(1 + \frac{\Delta p}{p_n \ln p_n}\right) = \frac{\Delta p}{p_n \ln p_n} \end{aligned}
In fact, \Delta p = p_{n+1} - p_n is called the "difference" of \{p_n\}. Previously, we used the derivative to approximate the difference, because the derivative \frac{dp}{dn} represents the slope of the tangent line at a point, while \frac{p_{n+1} - p_n}{(n+1) - n} is the average slope over the interval (n, n+1). In a smooth and monotonic graph, the two are approximate. Furthermore, since p_n and p_{n+1} are not far apart, we denote both as p, thus obtaining: \frac{dp}{dn} = \ln p
That is, n = \int \frac{1}{\ln p} dp. This is the strengthened version of the Prime Number Theorem! Unfortunately, this integral cannot be expressed as a combination of elementary functions, but this does not prevent us from using it. Mathematicians generally write it as: \pi(N) \sim Li(N) = \int_{0}^{N} \frac{1}{\ln t} dt
Like the Prime Number Theorem, it is correct; but as stated in Prime Obsession: It is more than true! This means it is an exceptionally good estimate. For the estimation of the number of primes within 10^{12}, the answer it provides has an error of only 0.0001%!
The above content is BoJone’s humble work for this week. Starting only from an approximate formula for the sum of the reciprocals of primes, we have obtained a profound theorem in modern number theory—the improved version of the Prime Number Theorem! This is the result of the comprehensive application of various mathematical techniques. There are reasonable parts containing a significant amount of "intuition," as well as some lack of rigor. Every step of the processing may not be strictly justified, but they are all quite interesting. Of course, this is not the reasoning process used by mathematicians, but rather a "brainstorming" session by the author, which I am happy to share and discuss with all readers!
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