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I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?

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- Generalized Alternating harmonic sum $\sum_{n\geq 1}\frac{\left(1-\frac{1}{2}+\frac{1}{3}-\cdots \pm \frac{1}{n}\right)}{n^p}$
- An arctan integral $\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx$
- An integral involving product of poly-logarithms and a power of a logarithm.
- Solve a problem on integration
- $\epsilon$-$\delta$ limit proof, $\lim_{x \to 2} \frac{x^{2}-2x+9}{x+1}$
- Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$
- Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

We have:

$$\begin{eqnarray*}2\left(\sum_{k=1}^{n}\sin(k^2)\right)^2 &=& \sum_{j,k=1}^{n}\cos(j^2-k^2)-\sum_{j,k=1}^{n}\cos(j^2+k^2)\\&=&n+2\sum_{m=1}^{n^2-1}d_1(m)\cos(m)-2\sum_{m=2}^{2n^2}d_2(m)\cos m\end{eqnarray*}$$

where $d_1(m)$ accounts for the number of ways to write $m$ as $j^2-k^2$ with $1\leq k<j\leq n$ and $d_2(m)$ accounts for the number of ways to write $m$ as $j^2+k^2$ with $1\leq j,k\leq n$. Since both these arithmetic functions do not deviate much from their average order (by Dirichlet’s hyperbola method $d_1(m)$ behaves on average like $\log m$ and $d_2(m)$ behaves on average like $\frac{\pi}{4}$), I think it is not difficult to prove that for infinitely many $n$s

$$ \left|\sum_{k=1}^{n}\sin(k^2)\right|\geq C\sqrt{n} $$

holds for some absolute constant $C\approx \frac{1}{\sqrt{2}}$ through the Cauchy-Schwarz inequality.

However, this is quite delicate: Weyl bounds work just in the opposite direction. As an alternative approach, we may consider only the set of $n$ that appear in the numerators of the convergents of $2\pi$, in order that our sum behaves like a Gaussian sum (with magnitude $\sqrt{n}$ or $\sqrt{2n}$) plus a small error. In both cases we get that the sequence given by

$$S_n=\sum_{k=1}^{n}\sin(k^2)$$

**is not** bounded.

In fact, Terence Tao already proved that on MO.

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