 Several different proofs of it were found, including the "elementary" proofs of Atle Selberg and Paul Erdős (1949).While the original proofs of Hadamard and de la Vallée Poussin are long and elaborate, later proofs introduced various simplifications through the use of Tauberian theorems but remained difficult to digest.A short proof was discovered in 1980 by American mathematician Donald J. Newman.