Convergence of $\sum \frac{\sin(n^2)}{n}$
Does the numerical series $\sum \frac{\sin(n^2)}{n}$ converges ?
For the moment I have tried a discrete integration by parts but it involves the asymptotic behaviour of $\sum \sin(n^2)$ which seems complex.
Trying a comparison with an integral does not seem very useful too.
By Weyl's inequality we have $$ \sum_{n=1}^{N}\sin(n^2)\ll N^{\frac{7}{12}} \log^2(N) $$ hence the given series is convergent by summation by parts.