Editing Analytic number theory (section)

===Hadamard and de la Vallée-Poussin===
{{main|Jacques Hadamard|Charles Jean de la Vallée-Poussin}}
Extending the ideas of Riemann, two proofs of the [[prime number theorem]] were obtained independently by [[Jacques Hadamard]] and [[Charles Jean de la Vallée-Poussin]] and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(''s'') is non-zero for all complex values of the variable ''s'' that have the form ''s''&nbsp;=&nbsp;1&nbsp;+&nbsp;''it'' with ''t''&nbsp;>&nbsp;0.<ref>{{cite book |last = Ingham |first = A.E. |title = The Distribution of Prime Numbers |publisher = Cambridge University Press| year = 1990 |pages = 2–5 |isbn = 0-521-39789-8}}</ref>