$$S = \sum_{k=1}^{\infty} \frac{\cos(\theta\log(k))}{k^a}$$
How do I go about finding the value of S, given that $\theta \to \infty$ and $0 < a < 1$.
Any special techniques that might be helpful in calculating this sum?
EDIT: Just to give some background,
I was actually trying to figure out $$\sum_{k=1}^{\infty} \frac{\cos(\theta\log(k))}{k^a} - \sum_{k=1}^{\infty} \frac{\cos(\theta\log(k + 0.5))}{(k+0.5)^a}$$
Since that expression was a bit complicated, I decided to write the common version...