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Given series $\sum_{n=1}^{\infty}\frac {\cos(n^2)}n$.

It is easy to prove it does not converge absolutely.

I need to prove that it converges сconditionally.

I thought about using Dirichlet's test because $1/n$ series is monotone and $\lim_{n\rightarrow\infty} 1/n = 0$.

So the thing i need to prove is $\left|\sum_{1}^k \cos(n^2)\right| < M,\ \forall k$.

If there was $\cos(n)$ instead of $n^2$, it would be easy to prove this statement by multiplication and division by $\cos(0.5)$ and then using some trigonometric formula so that $\left|\sum_{1}^k \cos(n^2)\right| = \left|\frac{cos(0.5)-cos(n-0.5)}{\cos(0.5)}\right| < 2$ or something like that. But this approach seems to be impossible for $\cos(n^2)$.

Maybe there is a way to prove it using the fact that $\int \cos(x^2) = \sqrt{2/\pi}$?

Or any simpler way?

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  • $\begingroup$ Your bounds would need to go from $1$ to $\infty$ since $\frac{\cos(n^2)}{n}$ is undefined at $0$. $\endgroup$
    – Kaj Hansen
    May 25, 2014 at 23:02
  • $\begingroup$ Sure, I'll fix it, thanks. $\endgroup$
    – Daniel
    May 25, 2014 at 23:03

1 Answer 1

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It is sufficient to show that $$\left|\sum_{n=0}^{N}\cos(n^2)\right|\leq C\sqrt{N}\log N\tag{1}$$ to ensure convergence by partial summation. Consider that: $$\left|\sum_{n=1}^{N}e^{in^2}\right|^2 = \left(\sum_{n=1}^{N}e^{in^2}\right)\cdot\left(\sum_{n=1}^{N}e^{-in^2}\right)=N+\sum_{d=1}^{N-1}\sum_{r=1}^{N-d}2\cos(2dr+d^2),\tag{2}$$ and that: $$(2)\ll \sum_{d=1}^{N-1}\min\left(N-d,\left\|\frac{d}{\pi}\right\|^{-1}\right)\tag{3}$$ (where $\|x\|$ denotes the distance of $x$ from the closest integer) by the usual arguments about simple exponential sums. If now we take $\frac{a}{q}$ as a good rational approximation of $\frac{1}{\pi}$, $\left|\frac{a}{q}-\frac{1}{\pi}\right|<\frac{1}{3Nq}$, it is not difficult to see that: $$ (3)\ll \sum_{\substack{d=1\\q\nmid d}}^{N}\left\|\frac{a d}{q}\right\|^{-1}\ll(N+q)\log q\ll N\log N,\tag{4}$$ hence $(1)$ holds and the series $\sum_{n=1}^{+\infty}\frac{\cos n^2}{n}$ converges.

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  • $\begingroup$ This is also known as "Weyl differencing trick". $(1)$ is just a consequence of the general en.wikipedia.org/wiki/Weyl%27s_inequality . $\endgroup$ May 25, 2014 at 23:31
  • $\begingroup$ I do not understand how to prove convergence then. What kind of convergence test that is? $\endgroup$
    – Daniel
    May 25, 2014 at 23:59
  • $\begingroup$ You can call it "generalized Dirichlet test", but is just partial summation. What we have is: $$\sum_{n=1}^{N}\frac{\cos(n^2)}{n}=\frac{1}{N}\sum_{n=1}^{N}\cos(n^2)+\sum_{n=1}^{N-1}\frac{1}{n(n+1)}\sum_{k=1}^{n}\cos(n^2)\ll\sum_{n=1}^{N}\frac{\sqrt{n}\log n}{n^2}.$$ $\endgroup$ May 26, 2014 at 2:14
  • $\begingroup$ Hi, great solution! I have a question: can you give some estimates for sums of form $\sum_{n=1}^N exp( i f(n))$ where $f$ is not a polynomial, but a bit more complicated, say $f(n) = \sqrt{n}$. ? $\endgroup$
    – orangeskid
    Aug 18, 2017 at 11:28
  • $\begingroup$ To anyone who is reading this after seven years, I think there is a (correctible) mistake in the proof. Namely, estimating the term $$\sum_{q | d} (N-d) \sim \frac{N^2}{2q}$$ requires a lower estimate for $q$, which depends on the Liouville-Roth constant. Luckily, we know that $\frac{1}{\pi}$ is not a Liouville number, so the series will still converge, but you will get some worse asymptotic growth than $O(\sqrt{N} \log N)$ for those cosines. $\endgroup$ May 22, 2021 at 8:22

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