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Ramanujan Function


$\displaystyle \phi(a,n)$ $\textstyle \equiv$ $\displaystyle 1+2\sum_{k=1}^n {1\over (ak)^3-ak}$  
$\displaystyle \phi(a)$ $\textstyle \equiv$ $\displaystyle \lim_{n\to\infty} \phi(a,n) = 1+2\sum_{k=1}^\infty {1\over (ak)^3-ak}.$  

The values of $\phi(n)$ for $n=2$, 3, ... are
$\displaystyle \phi(2)$ $\textstyle =$ $\displaystyle 2\ln 2$  
$\displaystyle \phi(3)$ $\textstyle =$ $\displaystyle \ln 3$  
$\displaystyle \phi(4)$ $\textstyle =$ $\displaystyle {\textstyle{3\over 2}} \ln 2$  
$\displaystyle \phi(6)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\ln 3+{\textstyle{2\over 3}}\ln 2.$  




© 1996-9 Eric W. Weisstein
1999-05-25