Let $x_{2}>x_{1}>0$ $$L(N)=\sum_{n=1}^{N}a_{n}(\frac{1}{n^{x_{1}}}+\frac{1}{n^{x_{2}}})$$ Where $a_{n}=\sum_{d|n}(-1)^{d+\frac{n}{d}}\cos(\ln(\frac{n}{d^{2}}))$
Problem:
Find $\inf_{h\ge 0}L(N+h)$ For given natural $N$
The things i done:
$\inf_{h\ge 0}L(N+h)=L(N)+\inf_{h\ge 0}\sum_{n=N+1}^{N+h}a_{n}(\frac{1}{n^{x_{1}}}+\frac{1}{n^{x_{2}}})$
I tried to use estymation, find lower bound for $M(N,h):=\sum_{n=N+1}^{N+h}a_{n}(\frac{1}{n^{x_{1}}}+\frac{1}{n^{x_{2}}})$
I am looking for such f that $M(N,h)=\theta(f(h))$
Any ideas?
Regards.