[Euler's Mathematics] The Prime Number Theorem and Its Refinement
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 $\pi(N)$ denotes the number of primes not exceeding N, and $\frac{N}{\ln N}$ is a computable expression; the symbol ~ means "is asymptotic to," so the whole formula reads "the number of primes not exceeding N is asymptotic to $\frac{N}{\ln N}$." In simple terms, $\frac{N}{\ln N}$ is an approximation of $\pi(N)$. Some readers may wonder why we don't just use ≈ instead of ~? In fact, ~ carries a stronger meaning:
$$\lim_{N-\infty} \frac{\pi(N) \ln N}{N}=1$$more
This is exactly the point of the ~ symbol — it also captures the limiting behavior, which ≈ does not require. Like the equals sign, ~ is transitive, i.e., if A~B~C then A~C. The Prime Number Theorem was proved in 1896 (using complex analysis), and later a number of other advanced or elementary proofs appeared. For details, see Wikipedia — Prime Number Theorem. The deep significance of the Prime Number Theorem lies in this: primes seem to appear with no discernible pattern, yet viewed as a whole, the count of primes turns out to obey a definite regularity — 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 itself — in other words, proving any one of them lets you derive the other two.
However, although the Prime Number Theorem is true, it is really quite crude: for estimating the number of primes below 10^12, it produces an error of 4%. Below, starting from the result of the previous article, we will derive a "strengthened" version of the Prime Number Theorem. In the previous article we obtained:
$$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{p}=Q > \ln \ln (p+1) -\ln2$$
and noted that $\ln \ln p$ is a rather good approximation to Q.
Let's make a rough estimate here, writing directly:
$$Q_n=\ln \ln p_n$$
and likewise
$$Q_{n+1}=Q_n+\frac{1}{p_{n+1}}=\ln \ln p_{n+1}$$
Subtracting the two gives
$$\frac{1}{p_{n+1}}=\ln \frac{\ln p_{n+1}}{\ln p_n}$$
that is, $\Delta p=p_{n+1}-p_n$. When $p_n$ is fairly large, $\Delta p$ is "negligible," and using the approximation $\ln(x+\varepsilon ) \approx \ln x+\frac{\varepsilon }{x}$, we can write:
$$\begin{aligned}\frac{1}{p_{n+1}}&=\ln \frac{\ln p_{n+1}}{\ln p_n}=\ln \frac{\ln p_n+\frac{\Delta p}{p_n}}{\ln p_n}\\ &=\ln(1+\frac{\Delta p}{p_n \ln p_n})=\frac{\Delta p}{p_n \ln p_n}\end{aligned}$$
In fact, $\Delta p=p_{n+1}-p_n$ is called the "difference" of ${p_n}$. We've previously used derivatives to approximate differences, because the meaning of the derivative $\frac{dp}{dn}$ is the slope of the tangent line at a point, whereas $\frac{p_{n+1}-p_n}{(n+1)-n}$ is the average slope over the interval (n, n+1); for smooth, monotonic curves the two are close. Moreover, since $p_n$ and $p_{n+1}$ differ very little, we denote both simply as p, and thus obtain
$$\frac{dp}{dn}=\ln p$$
that is, $n=\int \frac{1}{\ln p} dp$ — this is the strengthened Prime Number Theorem! Unfortunately, this integral cannot be expressed as a combination of elementary functions, but that doesn't stop us from using it. Mathematicians usually 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 Prime Obsession puts it: it holds even better than "holding true"! What this means is that it is an extraordinarily good estimate — for the number of primes below 10^12, the error it gives is only 0.0001%!
The above is BoJone's amateur work for this week. Starting merely from an approximation formula for the sum of reciprocals of primes, we arrived at a profound theorem in modern number theory — an improved version of the Prime Number Theorem! This is the result of combining various mathematical techniques. Some of the steps are reasonable and rely heavily on "intuition," while others admittedly lack full rigor. Not every step in the treatment is necessarily justified, but each one is, I think, quite interesting. Of course, this is not how mathematicians actually arrived at the result — it's merely the author's own "brainstorm," which I'd like to share and discuss with all readers!
**(Note: Internet Explorer causes the ~ symbol to render improperly, hence this remark.)*
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.