TITLE: Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$
QUESTION [26 upvotes]: $$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$

Prove it converges and,

evaluate the series.




For the first part of the question, I prove it converges by considering the irrationality measure,
$$|\sin (n\pi\sqrt{61})|=|\sin (n\pi\sqrt{61}-m\pi)|\ge \frac{2}{\pi}|n\pi\sqrt{61}-m\pi|\ge 2n\left|\sqrt{61}-\frac{m}{n}\right|>\frac{C}{n},$$
so
$$\left|\frac{\cot(\pi n\sqrt{61})}{n^3}\right|\le\left|\frac{1}{n^3\sin(\pi n\sqrt{61})}\right|<\frac{C}{n^2}.$$
How to evaluate the series?
I have found a related question.
I apologize for incorrect information in the previous post.

REPLY [8 votes]: Sigh. I believe that Pisco's sequence from $\zeta_1$ to $\zeta_{11}$ matches the middle column (all plus signs) in Prof. Lubin's no-calculator method for solving Pell, and is likely what Fermat, Wallis, Brouncker used all those years ago. Yes! If I continued one more step, I would get $\zeta_{11},$ which equals $\zeta_1$
Method described  by Prof. Lubin  at Continued fraction of $\sqrt{67} - 4$ 
$$  \sqrt { 61} = 7 +     \frac{  \sqrt {61} - 7 }{ 1 }  $$
$$    \frac{ 1 }{   \sqrt {61} - 7 }  =  \frac{    \sqrt {61} + 7 }{12 } = 1 +  \frac{    \sqrt {61} - 5 }{12 } $$
$$    \frac{ 12 }{   \sqrt {61} - 5 }  =  \frac{    \sqrt {61} + 5 }{3 } = 4 +  \frac{    \sqrt {61} - 7 }{3 } $$
$$    \frac{ 3 }{   \sqrt {61} - 7 }  =  \frac{    \sqrt {61} + 7 }{4 } = 3 +  \frac{    \sqrt {61} - 5 }{4 } $$
$$    \frac{ 4 }{   \sqrt {61} - 5 }  =  \frac{    \sqrt {61} + 5 }{9 } = 1 +  \frac{    \sqrt {61} - 4 }{9 } $$
$$    \frac{ 9 }{   \sqrt {61} - 4 }  =  \frac{    \sqrt {61} + 4 }{5 } = 2 +  \frac{    \sqrt {61} - 6 }{5 } $$
$$    \frac{ 5 }{   \sqrt {61} - 6 }  =  \frac{    \sqrt {61} + 6 }{5 } = 2 +  \frac{    \sqrt {61} - 4 }{5 } $$
$$    \frac{ 5 }{   \sqrt {61} - 4 }  =  \frac{    \sqrt {61} + 4 }{9 } = 1 +  \frac{    \sqrt {61} - 5 }{9 } $$
$$    \frac{ 9 }{   \sqrt {61} - 5 }  =  \frac{    \sqrt {61} + 5 }{4 } = 3 +  \frac{    \sqrt {61} - 7 }{4 } $$
$$    \frac{ 4 }{   \sqrt {61} - 7 }  =  \frac{    \sqrt {61} + 7 }{3 } = 4 +  \frac{    \sqrt {61} - 5 }{3 } $$
$$    \frac{ 3 }{   \sqrt {61} - 5 }  =  \frac{    \sqrt {61} + 5 }{12 } = 1 +  \frac{    \sqrt {61} - 7 }{12 } $$
$$    \frac{ 12 }{   \sqrt {61} - 7 }  =  \frac{    \sqrt {61} + 7 }{1 } = 14 +  \frac{    \sqrt {61} - 7 }{1 } $$ 
Simple continued fraction tableau:
$$ 
 \begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccc}
 & & 7 & & 1 & & 4 & & 3 & & 1 & & 2 & & 2 & & 1 & & 3 & & 4 & & 1 & & 14 & & 1 & & 4 & & 3 & & 1 & & 2 & & 2 & & 1 & & 3 & & 4 & & 1 & & 14 & \\ 
 \\ 
  \frac{ 0 }{ 1 }   &   \frac{ 1 }{ 0 }   & &   \frac{ 7 }{ 1 }   & &   \frac{ 8 }{ 1 }   & &   \frac{ 39 }{ 5 }   & &   \frac{ 125 }{ 16 }   & &   \frac{ 164 }{ 21 }   & &   \frac{ 453 }{ 58 }   & &   \frac{ 1070 }{ 137 }   & &   \frac{ 1523 }{ 195 }   & &   \frac{ 5639 }{ 722 }   & &   \frac{ 24079 }{ 3083 }   & &   \frac{ 29718 }{ 3805 }   & &   \frac{ 440131 }{ 56353 }   & &   \frac{ 469849 }{ 60158 }   & &   \frac{ 2319527 }{ 296985 }   & &   \frac{ 7428430 }{ 951113 }   & &   \frac{ 9747957 }{ 1248098 }   & &   \frac{ 26924344 }{ 3447309 }   & &   \frac{ 63596645 }{ 8142716 }   & &   \frac{ 90520989 }{ 11590025 }   & &   \frac{ 335159612 }{ 42912791 }   & &   \frac{ 1431159437 }{ 183241189 }   & &   \frac{ 1766319049 }{ 226153980 }    \\ 
  \\ 
 & 1 & & -12 & & 3 & & -4 & & 9 & & -5 & & 5 & & -9 & & 4 & & -3 & & 12 & & -1 & & 12 & & -3 & & 4 & & -9 & & 5 & & -5 & & 9 & & -4 & & 3 & & -12 & & 1
 \end{array}
 $$ 
$$ 
 \begin{array}{cccc}
  \frac{ 1 }{ 0 }   & 1^2 - 61 \cdot 0^2 = 1 &     \mbox{digit}  &  7  \\  
  \frac{ 7 }{ 1 }   & 7^2 - 61 \cdot 1^2 = -12 &     \mbox{digit}  &  1  \\  
  \frac{ 8 }{ 1 }   & 8^2 - 61 \cdot 1^2 = 3 &     \mbox{digit}  &  4  \\  
  \frac{ 39 }{ 5 }   & 39^2 - 61 \cdot 5^2 = -4 &     \mbox{digit}  &  3  \\  
  \frac{ 125 }{ 16 }   & 125^2 - 61 \cdot 16^2 = 9 &     \mbox{digit}  &  1  \\  
  \frac{ 164 }{ 21 }   & 164^2 - 61 \cdot 21^2 = -5 &     \mbox{digit}  &  2  \\  
  \frac{ 453 }{ 58 }   & 453^2 - 61 \cdot 58^2 = 5 &     \mbox{digit}  &  2  \\  
  \frac{ 1070 }{ 137 }   & 1070^2 - 61 \cdot 137^2 = -9 &     \mbox{digit}  &  1  \\  
  \frac{ 1523 }{ 195 }   & 1523^2 - 61 \cdot 195^2 = 4 &     \mbox{digit}  &  3  \\  
  \frac{ 5639 }{ 722 }   & 5639^2 - 61 \cdot 722^2 = -3 &     \mbox{digit}  &  4  \\  
  \frac{ 24079 }{ 3083 }   & 24079^2 - 61 \cdot 3083^2 = 12 &     \mbox{digit}  &  1  \\  
  \frac{ 29718 }{ 3805 }   & 29718^2 - 61 \cdot 3805^2 = -1 &     \mbox{digit}  &  14  \\  
  \frac{ 440131 }{ 56353 }   & 440131^2 - 61 \cdot 56353^2 = 12 &     \mbox{digit}  &  1  \\  
  \frac{ 469849 }{ 60158 }   & 469849^2 - 61 \cdot 60158^2 = -3 &     \mbox{digit}  &  4  \\  
  \frac{ 2319527 }{ 296985 }   & 2319527^2 - 61 \cdot 296985^2 = 4 &     \mbox{digit}  &  3  \\  
  \frac{ 7428430 }{ 951113 }   & 7428430^2 - 61 \cdot 951113^2 = -9 &     \mbox{digit}  &  1  \\  
  \frac{ 9747957 }{ 1248098 }   & 9747957^2 - 61 \cdot 1248098^2 = 5 &     \mbox{digit}  &  2  \\  
  \frac{ 26924344 }{ 3447309 }   & 26924344^2 - 61 \cdot 3447309^2 = -5 &     \mbox{digit}  &  2  \\  
  \frac{ 63596645 }{ 8142716 }   & 63596645^2 - 61 \cdot 8142716^2 = 9 &     \mbox{digit}  &  1  \\  
  \frac{ 90520989 }{ 11590025 }   & 90520989^2 - 61 \cdot 11590025^2 = -4 &     \mbox{digit}  &  3  \\  
  \frac{ 335159612 }{ 42912791 }   & 335159612^2 - 61 \cdot 42912791^2 = 3 &     \mbox{digit}  &  4  \\  
  \frac{ 1431159437 }{ 183241189 }   & 1431159437^2 - 61 \cdot 183241189^2 = -12 &     \mbox{digit}  &  1  \\  
  \frac{ 1766319049 }{ 226153980 }   & 1766319049^2 - 61 \cdot 226153980^2 = 1 &     \mbox{digit}  &  14  \\  
 \end{array}
 $$