We are investigating a problem that looks simple but requires deep machinery to solve perfectly. The goal is to prove:
$$ \lim_{x \to \infty} \frac{\pi(x) \ln(x)}{x} = 1 $$where $\pi(x)$ is the number of primes less than or equal to $x$.
The standard strategy in calculus is the Squeeze Theorem. Ideally, we want to find a Lower Bound $L(x)$ and an Upper Bound $U(x)$ such that $ L(x) \le \pi(x) \le U(x) $.
If we can force both bounds to the same limit, the chaotic, jagged step-function of the primes is forced to comply.
Proving the limit is exactly $1$ requires complex analysis. However, if we accept a looser limit, we can use elementary algebra (Pascal's Triangle).
By analyzing the central binomial coefficient $\binom{2n}{n}$, Chebyshev proved we can bound the primes with purely algebraic tools. In the plot above (click "The Easy Way"), you can see this mathematically tractable bound:
$$ \ln(2) \frac{x}{\ln x} < \pi(x) < \ln(4) \frac{x}{\ln x} $$This traps the ratio between 0.69 and 1.38. It proves the ratio is finite and bounded without needing imaginary numbers.
To close the gap from 1.38 down to exactly 1.00, algebra was not enough. The barrier was broken independently in 1896 by Jacques Hadamard and Charles Jean de la Vallée Poussin.
They bridged the gap by proving a profound fact about the Riemann Zeta Function $\zeta(s)$: It has no zeros where the real part is 1.
This single fact forces the error term to vanish, turning the approximation into an equality at infinity:
$$ \pi(x) \approx \text{Li}(x) - \sum_{\rho} \text{Li}(x^{\rho}) $$"By proving that $\zeta(1+it) \neq 0$, they ensured that the music of the primes contains no dissonance that would disrupt the limit."