There is a sequence: $$a_{n}=\sum_{d|n}(-1)^{d+\frac{n}{d}}\cos(\ln(\frac{n}{d^{2}}))$$
Is this true that $a_{n}\rightarrow0$ ?
It seems that this sequence rather diverges.
I tried to prove its discrepancy by looking for special divergent subsequences, but i can't find.
I hope that someone will help me.
Regards.
If $p$ is an odd prime, then $$a_p=(-1)^{p+1}(\cos(\ln(p))+\cos(\ln(1/p))=2\cos(\ln(p)).$$ By Bertrand's postulate, the logarithms of consecutive primes are within $\ln(2)<\pi/4$ of each other. It follows that every interval of the form $[N\pi/4,(N+1)\pi/4]$ for $N$ a positive integer contains a number of the form $\ln(p)$ for some prime $p$. In particular, this means that $\cos(\ln(p))$ takes both values greater than $1/\sqrt{2}$ and values less than $-1/\sqrt{2}$ infinitely often. It follows that the sequence $(a_p)$ cannot converge as $p$ ranges over the primes.