The prime counting function π(x) counts primes up to x. Chebyshev proved bounds c₁x/ln(x) < π(x) < c₂x/ln(x) for explicit constants, providing quantitative control on prime density and laying essential groundwork for the Prime Number Theorem.
You already know from the distribution of primes that primes thin out as numbers grow: the gaps between consecutive primes generally increase, and heuristically there are about x/ln(x) primes up to x. The prime counting function π(x) makes this precise — it is the exact count of primes p ≤ x. For example, π(10) = 4 (the primes 2, 3, 5, 7), π(100) = 25, and π(1,000,000) = 78,498. The central question of analytic number theory is: how does π(x) grow asymptotically?
Chebyshev's contribution was to establish rigorous bounds without proving the exact asymptotic. He introduced two auxiliary functions: θ(x) = Σ_{p ≤ x} ln(p) (summing logarithms over primes) and ψ(x) = Σ_{n ≤ x} Λ(n) (using the von Mangoldt function). These are smoother and more tractable analytically than π(x) directly. The key proof strategy analyzes the central binomial coefficient C(2n, n) = (2n)!/(n!)², which is easy to bound: 4^n/(2n) < C(2n, n) < 4^n by elementary means. The prime factorization of C(2n, n) gives information about θ, and iterating these bounds yields explicit constants c₁ ≈ 0.92 and c₂ ≈ 1.11 such that c₁x/ln(x) < π(x) < c₂x/ln(x) for all sufficiently large x.
Chebyshev's result is foundational for two reasons. Practically, the bounds show that if π(x)/(x/ln x) has any limit at all, that limit must be 1 — there is no room for a different leading constant. Structurally, the tools Chebyshev introduced (θ and ψ, their relationship to π via Abel summation, the connection to the central binomial coefficient) are exactly the tools that Riemann, Hadamard, and de la Vallée Poussin would refine to prove π(x) ~ x/ln(x) — the Prime Number Theorem. Chebyshev did not reach the summit, but he built the base camp and established that the summit exists at exactly the right altitude.