I came across this problem while doing some simplifications.


So, I like to ask



> 
> **QUESTION.** Is there a closed formula for the evaluation of this series?
> $$\sum\_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{a^2b^2}$$
> where the sum runs over all pairs of positive integers that are relatively prime.
> 
> 
> 



Closed formula for a certain infinite series
Apply Möbius summation, the formula for $\sum\_{n>=1}\cos(2\pi n x)/n^2$ to obtain:


$$11/4-45\zeta(3)/\pi^3=1.00543...\;$$


