Tate’s Thesis and Applications

Abstract

It is well known that much information on rational primes is encoded in the Riemann zeta function ζ(s), which is defined by the absolutely convergent series

EquationSource % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOWdyLaai % ikaiaadohacaGGPaGaeyypa0ZaaabuaeaadaWcaaqaaiaaigdaaeaa % caWGUbWaaWbaaSqabeaacaWGZbaaaaaaaeaacaWGUbGaeyyzImRaaG % ymaaqab0GaeyyeIuoaaaa!4358! \[ \varsigma (s) = \sum\limits_{n \geqslant 1} {\frac{1} {{n^s }}} \

for complex numbers s such that Re(s)>1.