theorems have been proved under the assumption that the hypothesis is true. For example, under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is In the early 20th century G. H. Hardy and Littlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis. In fact, in 1914, Hardy proved that there were infinitely many zeros of the zeta function on the critical line This led to several theorems describing the density of the zeros on the critical line. See also Automorphic L-function Automorphic form Langlands program Maier's matrix method Notes References Further reading Ayoub, Introduction to the Analytic Theory of Numbers H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I : Classical Theory H. Iwaniec and E. Kowalski, Analytic Number Theory. D. J. Newman, Analytic number theory, Springer, 1998 On specialized aspects the following books have become especially well-known: H. Halberstam and H. E. Richert, Sieve Methods R. C. Vaughan, The Hardy–Littlewood method, 2nd. edn. Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.