Prime number theorem

The prime number theorem states that the number of primes at most x is asymptotic to x divided by the natural logarithm of x. It quantifies the long-run density of primes.

Prime-counting function: The function π(x) counts the prime numbers less than or equal to x. The theorem gives an asymptotic estimate for this counting function.
Riemann zeta function: The complex function defined by the analytically continued series ζ(s) = Σ n⁻ˢ for real part of s greater than one. Its zeros encode fluctuations in prime counting, making it central to the analytic proof.
Bernhard Riemann: A German mathematician whose work on the zeta function transformed the study of prime distribution. His 1859 memoir supplied the analytic framework that led toward the theorem.
Asymptotic analysis: The study of the limiting behavior of functions as their arguments approach a specified value or infinity. The theorem compares prime counts with a simpler function as x tends to infinity.
Prime gaps: The differences between consecutive prime numbers. The theorem implies that the average gap near x is on the scale of log x.
Prime number theorem for arithmetic progressions: A theorem describing prime distribution among residue classes coprime to a fixed modulus. It extends the theorem's density prediction to primes in specified congruence classes.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. The prime number theorem is known without this stronger claim, which would sharpen its error estimates.
Jacques Hadamard: A French mathematician who proved the prime number theorem in 1896. His proof used complex analysis to establish the theorem from zeta-function properties.
Natural logarithm: The logarithm to base e, the inverse function of the exponential function. Its growth rate sets the scale of the theorem's estimate for prime density.
Prime density: A measure of how frequently primes occur among positive integers or within a specified set. The theorem says the proportion of primes up to x is asymptotic to 1/log x.

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