• D Zagier

      Articles written in Proceedings – Mathematical Sciences

    • On an approximate identity of Ramanujan

      D Zagier

      More Details Abstract Fulltext PDF

      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.

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