• On an approximate identity of Ramanujan

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      https://www.ias.ac.in/article/fulltext/pmsc/097/01-03/0313-0324

    • Keywords

       

      Approximate identity; asymptotic behaviour; Rogers—Ramanujan identities; dilogarithm function

    • Abstract

       

      In his second notebook, Ramanujan says that$$\frac{q}{{x + }}\frac{{q^4 }}{{x + }}\frac{{q^8 }}{{x + }}\frac{{q^{12} }}{{x + }} \cdots = 1 - \frac{{qx}}{{1 + }}\frac{{q^2 }}{{1 - }}\frac{{q^3 x}}{{1 + }}\frac{{q^4 }}{{1 - }} \cdots $$ “nearly” forq andx between 0 and 1. It is shown in what senses this is true. In particular, asq → 1 the difference between the left and right sides is approximately exp −c(x)/(l-q) wherec(x) is a function expressible in terms of the dilogarithm and which is monotone decreasing with c(0) = π2/4,c(1) = π2/5; thus the difference in question is less than 2· l0−85 forq = 0·99 and allx between 0 and 1.

    • Author Affiliations

       

      D Zagier1 2

      1. Max-Planck Institut für Mathematik, Gottfried-Claren-Straße 26, Bonn - D-5300, FRG
      2. Department of Mathematics, University of Maryland, College Park, Maryland - 20742, USA
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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