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Ramanujan-Eisenstein Series

Let $t$ be a discriminant,

q\equiv -e^{-\pi\sqrt{t}},
\end{displaymath} (1)

$\displaystyle E_2(q)$ $\textstyle \equiv$ $\displaystyle L(q)\equiv 1-24\sum_{k=1}^\infty {(2k+1)q^{2k+1}\over 1-q^{2k+1}}$  
  $\textstyle =$ $\displaystyle \left({2K\over \pi}\right)^2(1-2k^2)$ (2)
$\displaystyle E_4(q)$ $\textstyle \equiv$ $\displaystyle M(q)\equiv 1+240\sum_{k=1}^\infty {k^3q^{2k}\over 1-q^{2k}}$  
  $\textstyle =$ $\displaystyle \left({2K\over \pi}\right)^4(1-k^2k'^2)$ (3)
$\displaystyle E_6(q)$ $\textstyle \equiv$ $\displaystyle N(q)\equiv 1-504\sum_{k=1}^\infty {k^5q^{2k}\over 1-q^{2k}}$  
  $\textstyle =$ $\displaystyle \left({2K\over \pi}\right)^6(1-2k^2)(1+{\textstyle{1\over 2}}k^2k'^2).$ (4)

See also Klein's Absolute Invariant, Pi


Borwein, J. M. and Borwein, P. B. ``Class Number Three Ramanujan Type Series for $1/\pi$.'' J. Comput. Appl. Math. 46, 281-290, 1993.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.

© 1996-9 Eric W. Weisstein