Évènement

Abstract : In 1896, Hadamard and de La Vallée Poussin proved the famous prime number theorem, which gives an asymptotic formula in $x$ for the number of prime numbers $p\leq x$. Their proof heavily relies on complex analysis. Hardy, along with some other mathematicians in the 20\textsuperscript{th} century, wondered whether or not some theorems could only be proved using integers/real numbers/complex numbers. Hardy believed that the prime number theorem was one of those that required the use of complex numbers. This turned out not to be the case, as was discovered when Selberg and Erdös proved the prime number theorem using only elementary means. That being said, why is it natural to consider complex numbers when proving some arithmetic statements?

The aim of this talk is to outline the ideas that link prime numbers to deep tools such as exponential sums and the residue theorem. Of course, we will have to take some time to see how the Riemann Hypothesis relates to all these facts.