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Calculus Level 5

Definition:

\[\displaystyle \text{Li}_{s_1,s_2,\dots,s_k}(z_1,z_2,\dots,z_k)=\sum_{n_1>n_2>\dots>n_k>0}\prod_{j=1}^{k}n_j^{-s_j}z_j^{n_j}\]

Multiple polylogarihtms are multiply nested sums of the form above.


\[\displaystyle \text{Li}_{3,1,1}(-1)=\frac{\zeta(A)}{B}-\zeta(C)(\frac{\pi^D}{E}+\frac{F}{G}\log^H I)-J\text{Li}_K(\frac{L}{M})-J\text{Li}_N(\frac{L}{M})\log I+\frac{\pi^O\log^P Q}{R}-\frac{\log^S T}{U}\]

In the above equation, \(A,B,C,\ldots,U\) are positive integers. Hence, find the minimum value of:

\[ A+B+C+D+E+F+G+H+I+J \\ +K+L+M+N+O+P+Q+R+S+T+U. \]

Details and Assumptions:

  • \(\text{Li}_{3,1,1}(-1)\) is a multi polylogarithm.
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