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Calculus Level 5

Definition:

Lis1,s2,,sk(z1,z2,,zk)=n1>n2>>nk>0j=1knjsjzjnj\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.


Li3,1,1(1)=ζ(A)Bζ(C)(πDE+FGlogHI)JLiK(LM)JLiN(LM)logI+πOlogPQRlogSTU\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,,UA,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. A+B+C+D+E+F+G+H+I+J \\ +K+L+M+N+O+P+Q+R+S+T+U.

Details and Assumptions:

  • Li3,1,1(1)\text{Li}_{3,1,1}(-1) is a multi polylogarithm.
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