BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Wadim Zudilin DTSTART:20200914T160000Z DTEND:20200914T170000Z DTSTAMP:20260423T035918Z UID:UNYONTC/9 DESCRIPTION:Title: Creative microscoping\nby Wadim Zudilin as part of Upstate New York On line Number Theory Colloquium\n\n\nAbstract\nLet $A_k=2^{-6k}{\\binom{2k}k }^3$ for $k=0\,1\,\\dots$\,.\nThough traditional techniques of establishin g the hypergeometric evaluation\n\n$\\sum \\limits_{k=0}^\\infty(-1)^k(4k+ 1)A_k =\\frac2\\pi$\n\nand (super)congruences\n\n$\\sum \\limits_{k=0}^{p- 1}(-1)^k(4k+1)A_k \\equiv p(-1)^{(p-1)/2}\\pmod{p^3} \\quad\\text{for\npri mes}\\\; p>2$\n\nshare certain similarities\, they do not display intrinsi c reasons for the two to be\nrelated.\nIn my talk I will outline basic ing redients of a method developed in joint works\nwith Victor Guo\, which doe s the missing part\, also for many other instances of such\narithmetic dua lity.\nThe main idea is constructing suitable $q$-deformations of the infi nite sum (and\nmany such sums are already recorded in the $q$-literature)\ ,\nand then look at the asymptotics of that at roots of unity.\nInterestin gly enough\, the $q$-deformations may offer more.\nFor example\, the $q$-d eformation of the above infinite sum also implies\n$$\n\\sum_{k=0}^\\infty A_k\n=\\frac{\\Gamma(1/4)^4}{4\\pi^3}\n=\\frac{8L(f\,1)}{\\pi}\n\\quad\\t ext{and}\\quad\n\\sum_{k=0}^{p-1}A_k\\equiv a(p)\\pmod{p^2}\n$$\n(in fact\ , the latter congruences in their stronger modulo $p^3$ form proven by Lon g\nand Ramakrishna)\,\nwhere $a(p)$ is the $p$-th Fourier coefficient of ( the weight 3 modular form)\n$f=q\\prod_{m=1}^\\infty(1-q^{4m})^6$.\n\n($NB :$ The variable $q$ in the last definition is related to the modular\npara meter $\\tau$ through $q=e^{2\\pi i\\tau}$ and has nothing to do with the $q$ in\nthe $q$-deformation.)\n LOCATION:https://researchseminars.org/talk/UNYONTC/9/ END:VEVENT END:VCALENDAR