Prime-counting function

The function π(x) counts the prime numbers less than or equal to x. Its growth describes how primes are distributed among the integers.

Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. These are precisely the numbers counted by π(x).
Prime gap: The difference between consecutive prime numbers. Changes in π(x) encode how many primes fit between successive values.
Euclid's theorem: The theorem that infinitely many prime numbers exist. An unbounded count becomes meaningful because primes never run out.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. It would tightly constrain the error between π(x) and its main approximations.
Prime-counting function at powers of ten: The values of π(x) at x equal to powers of ten, recording prime counts across successive scales. These tabulated values make the function’s growth concrete.
Bertrand's postulate: The theorem that for every integer n greater than 1, a prime lies between n and 2n. It guarantees a minimum increase in prime counts across doubled intervals.
Leonhard Euler: An eighteenth-century mathematician whose work connected prime numbers with infinite series and products. Euler’s prime-product identity helped make prime distribution an analytic subject.
Prime-counting error term: The difference between the prime-counting function and a chosen approximation to it. Its size and oscillation measure how irregularly primes depart from average density.
Prime number theorem: A theorem stating that the number of primes up to x is asymptotic to x divided by its natural logarithm. It gives the standard large-x approximation π(x) ≈ x/log x.
Rosser's theorem: A theorem giving explicit upper and lower bounds for the nth prime in terms of n. Bounds on the nth prime translate into bounds on π(x).

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