Beyond Basel

Algebra Level 4

There exist positive integers A,B,C,D,E,FA,B,C,D,E,F such that p=1ncot6(pπ2n+1)=1Fn(2n1)(An4+Bn3+Cn2Dn+E) \sum_{p=1}^n \cot^6 \left(\tfrac{p \pi}{2n+1}\right) = \tfrac{1}{F}n(2n-1)\big(An^4 + Bn^3 + Cn^2 - Dn + E\big) for all positive integers nn, where FF is as small as possible.

What is A+B+C+D+E+F?A + B + C + D + E + F?


Bonus: Use this result to prove that p=11p6=1945π6. \sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6.

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