 The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the formula_1-th prime is known.Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
with relatively prime integers formula_34 and formula_39 take infinitely many prime values.Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same formula_39 have approximately the same proportions of primes.