BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Siddhi Pathak (Pennsylvania State University) DTSTART:20200529T123000Z DTEND:20200529T133000Z DTSTAMP:20260404T145131Z UID:CMI/9 DESCRIPTION:Title: Arithmetic nature of special values of L-functions\nby Siddhi Patha k (Pennsylvania State University) as part of CMI seminar series\n\n\nAbstr act\nThe study of L-functions has occupied a center stage in number theory since the work of Riemann and Dirichlet. A standard example of an L-funct ion is the Riemann zeta-function\, $\\zeta(s)$\, given by the series $\\s um_{n=1}^{\\infty} n^{-s}$ when $\\Re(s)>1$. The aim of this talk will be to discuss the question of determining the arithmetic nature (that is\, ra tional/irrational and algebraic/transcendental) of values of L-functions a t positive integers. For example\, it is expected that the values $\\zeta( m)$ are transcendental for all integers $m >1$. However\, the only known c ases of this conjecture are the even zeta-values\, which Euler had explici tly evaluated in the 1730s. Among the odd zeta-values\, Apery proved that $\\zeta(3)$ is irrational\, whereas the irrationality of the remaining odd zeta-values remains a mystery. \n\nIn this talk\, we will discuss various facets of this problem. If time permits\, we will prove that for a fixed odd positive integer m\, all the values $\\zeta_K(m)$ are irrational as K varies over imaginary quadratic fields\, with at most one possible excepti on. This is joint work with Ram Murty. This talk will be accessible to a w ide audience.\n LOCATION:https://stable.researchseminars.org/talk/CMI/9/ END:VEVENT END:VCALENDAR