The Hadamard and de la Vallée Poussin  proofs used the zeta function and were followed by a J.E. Littlewood proof that this theorem is not an inequality that used the Riemann hypothesis.
Hadamard and de la Vallée Poussin independently proved this theorem in 1896, and a proof of this involving only real numbers led to a dispute between Selberg and Erdős (AIR-dawsh). The Hadamard and de la Vallée Poussin  proofs used the zeta function and were followed by a J.E. Littlewood proof that this theorem is not an inequality that used the Riemann hypothesis. A concise way to state this theorem is that pi of n equals the logarithmic integral of n, where the logarithmic integral is d x over the natural log of x from two to n, which is approximately n over the natural log of n. Name this theorem which approximates how many numbers less than a given number have only one proper divisor.
Hadamard and de la Vallée Poussin independently proved this theorem in 1896, and a proof of this involving only real numbers led to a dispute between Selberg and Erdős (AIR-dawsh).
prime number theorem