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Chebyshev's bounds show π(x) is of order x/ln(x), proving primes have asymptotic density 1/ln(x). These classical bounds can be proved using elementary methods (Bertrand's postulate) without complex analysis, providing quantitative control on prime distribution.
You already know from the prime counting function π(x) that primes thin out as numbers grow — there are fewer primes near a million than near ten. Chebyshev's achievement was to make this vague observation precise: he proved that π(x) is trapped between two constant multiples of x/ln(x). In other words, the fraction of numbers up to x that are prime is approximately 1/ln(x), and this estimate is off by at most a bounded factor. This was the first quantitative result about prime density, proved without any complex analysis.
The central tool is a clever counting argument using the central binomial coefficient C(2n, n). This number is divisible by every prime between n and 2n (since each such prime divides the numerator (2n)! but not the denominator (n!)²). At the same time, C(2n, n) ≤ 4n because it is one term in the expansion of (1+1)2n. By tracking which primes divide C(2n, n) and bounding from above and below, Chebyshev extracted the bound θ(x) ~ x, where θ(x) = Σ ln(p) over primes p ≤ x is the Chebyshev theta function. From this, the upper and lower bounds on π(x) follow by partial summation.
Bertrand's postulate — that there is always a prime between n and 2n — enters as both a consequence and a proof tool. The argument is essentially self-reinforcing: the existence of primes in each doubling interval [n, 2n] guarantees that C(2n, n) is large enough to force the prime count upward. Chebyshev used this to establish 0.92 · x/ln(x) < π(x) < 1.105 · x/ln(x) for large x, explicit constants that had never been achieved before.
What Chebyshev could not prove — and what remained open until 1896 — is that the limiting ratio π(x)/(x/ln(x)) exists and equals exactly 1. That sharper statement is the Prime Number Theorem, proved by Hadamard and de la Vallée Poussin using the Riemann zeta function and complex analysis. Chebyshev's bounds sit just below that threshold: they prove the right order of magnitude without pinning down the exact constant. The gap between Chebyshev's elementary methods and the full PNT illustrates a general principle in analytic number theory — getting the leading term often requires deeper analytic machinery than getting the correct magnitude.