Divergence of $\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$, where $\gamma$ is the Euler-Mascheroni constant

We denote the Möbius function as $\mu(n)$, see its definition from this MathWorld.

On the other hand it is not known if the Euler-Mascheroni constant is irrational.

After I've read a MathOverflow post I would like to ask a related question.

Question. Is feasible to deduce if $$\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right),\tag{1}$$ is divergent? I am asking about what work can be done. Many thanks.

If you want to add a some words or a draft about if our series can be summable by any Cesàro means it also is good.

• I don't see any hope with current technology. We don't know whether $\gamma$ is irrational. With $n^2$ replaced by $n$, it is still difficult. – Sungjin Kim Mar 1 '18 at 22:45
• I recommend you reading this MO post: mathoverflow.net/questions/164874/… which gives divergence of $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt n}$. – Sungjin Kim Mar 1 '18 at 22:47
• I saw the answer from the MathOverflow's post I my thoughts after I did a graph of the partial sums $\sum_{n=1}^N\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$ for some $N's$, is that maybe someone can do some work about it. I accept your words and many thanks for your great reference @i707107 – user243301 Mar 1 '18 at 22:51