$\sum\limits_{k=1}^{\infty}\dfrac{1}{k^{1+|\sin k|}}$ has been shown to diverge. This made me wonder about the asymptotics of $S(n)=\sum\limits_{k=1}^n\dfrac{1}{k^{1+|\sin k|}}$.
Numerical investigation suggests that $S(n)$ against $\ln (\ln n)$ may have a linear asymptote:
Does $L=\lim\limits_{n\to\infty}\dfrac{\sum\limits_{k=1}^n {k^{-(1+|\sin k|)}}}{\ln (\ln n)}$ equal a positive real number, and if so, does it have a closed form?