The series $\sum\_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum\_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)


I came across the problem of showing that $\sum\_{n\geq 1} \frac{|\sin (n^2)|}{n}$ is divergent in [convergence of $\sum\_{n\geq 1}|\sin(n^2)|/{n}$](https://math.stackexchange.com/questions/1901485/convergence-of-sum-n-frac-sinn2n/1901539). A natural question is if it converges conditionally without the absolute signs. 


More generally, if given a polynomial $p(n)$ where the dominant coefficient is rationally independent from $\pi$ then by Weyl, see e.g. [Tao-Weyl equidistribution](https://terrytao.wordpress.com/2010/03/28/254b-notes-1-equidistribution-of-polynomial-sequences-in-torii/) $p(n)$ is equidistributed (mod 1) but is the equidistribution sufficiently rapid for the series $\sum\_{n\geq 1} \sin(p(n))/n$ to be conditionally convergent?



Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?
This is something I learned from fedja (artofproblemsolving.com):


We will show that if $x$ is not Liouvillian then the sum 
$$
\sum\_{n \geq 1} \frac{\sin (2 \pi x n^{2})}{n}
$$
converges. 
An argument goes like this:
rewrite 
$$
\sum\_{k =1}^{n} \frac{\sin (2 \pi x k^{2})}{k} = \frac{1}{n}\sum\_{k=1}^{n}\sin (2 \pi x k^{2}) + \sum\_{k=1}^{n} \frac{S\_{k}}{k(k+1)}=I+II
$$
where $S\_{k} = \sum\_{j=1}^{k}\sin (2 \pi x k^{2})$.
The first term is OK (converges to integral). For the second term it is enough to show that $|S\_{k}| = O(k^{\delta})$ for some $0<\delta<1$. We use the following trick:


$$
S\_{n}^{2} \leq \left| \sum\_{k=0}^{n} e^{2 \pi i x k^{2}}\right|^{2} = \sum\_{k,m} e^{2\pi i x(k-m)(k+m)} = (\*)
$$
Next change variables $k-m=h$ and $m=m$ then 


$$
(\*) =\sum\_{-n \leq h \leq n}e^{2 \pi i x h^{2}} \sum\_{1\leq m \leq n, 1 \leq m+h \leq n}e^{2 \pi i \cdot 2hxm} \leq \sum\_{-n \leq h \leq n}\left| \sum\_{1\leq m \leq n, 1 \leq m+h \leq n}e^{2 \pi i \cdot 2hxm}\right|
$$
Clearly (Dirichlet kernel)


$$
\left| \sum\_{1\leq m \leq n, 1 \leq m+h \leq n}e^{2 \pi im \cdot 2hx}\right| \leq \frac{C}{\|2hx\|}
$$
where $\|2hx\| = |2hx \mod 1|$ is the distance to the nearest integer. Since $x$ is not Liouvillian then there exists $0<q\leq n$ such that $|2x - \frac{p}{q}| \leq \frac{1}{nq}$ and $q>n^{a}$ for some $a>0$. This means that when $h \in [-n, n]$ then $2hx$ is close to the fraction of the type $h p /q$ up to an error $|h/qn|\leq 1/n^{a}$ therefore when $h$ runs over $[-n, n]$ then $2hx \mod 1$ almost runs over the fractions $\ell/q$, $\ell <q/2$ and each of them may appear at most $10 n/q$ times. Therefore
$$
\sum\_{-n \leq h \leq n} \frac{C}{\|2hx\|}\leq C\_{2} \frac{n}{q}\left( n+\sum\_{\ell=1}^{q/2}\frac{q}{\ell}\right)
$$ 


(The first $n$ in the parenthesis comes from the error). And everything is bounded as $C\_{3}( n^{2}/q+n\ln q) \leq C\_{3} (n^{2-a}+n\ln n)$. 


