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for all complex values of the variable that have the form with

During the 20th century, the theorem of Hadamard and de la Vallée Poussin also became known as the Prime Number Theorem. Several different proofs of it were found, including the "elementary" proofs of Atle Selberg (1949) and Paul Erdos (1949). Hadamard's and original proofs are long and elaborate; later proofs introduced various simplifications through the use of Tauberian theorems

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