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I recently stumbled over the statement on Facebook, but it was without a proof and I wasn't able to find one myself for a while now, and also asked around at the university without results. Plots I did for the partial sums showed pretty strange oscillating behavior that made it plausible to me that the series might be unbounded.

My current approach to proving it was to describe the m-th partial sum as $s_n= \Re t_m$ with $t_m := \sum_{j=0}^{m} \exp(ij^2)$, substituting in $j^2 = \sum_{k=0}^{j-1}2k+1$ and trying to find a second recursive equation like you would do it for the partial sums of the geometric series, but I didn't find anything that would've helped me.

Do you guys already know this series or have an idea?

Jack D'Aurizio
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    By Weyl's inequality (https://en.wikipedia.org/wiki/Weyl's_inequality) we have $$ \left|s_m\right| \leq C \sqrt{m}\log^2(m) $$ for large $m$s. The "tightness" of such inequality depends on the irrationality measure of $\pi$. We may probably prove through Van Der Corput's trick and Sarkozy's theorem (https://arxiv.org/abs/1107.0243) that $$\left|s_m\right|\geq D m^{1/10} $$ holds for infinite $m$s. – Jack D'Aurizio Jan 26 '17 at 20:46
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    Here it is a proof that $s_m$ is unbounded: http://mathoverflow.net/questions/201250/is-sum-k-1n-sink2-bounded-by-a-constant-m – Jack D'Aurizio Jan 26 '17 at 20:50

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