You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real parts, due to $e^{2\pi i a}=\cos(2\pi a)+i\sin(2\pi a)$. Now when you plug in a root of $M(n)$ (I listed some here), you can easily see that the imaginary part is zero since the Farey sequence is symmetric around $\frac12$.
Further when you take $n$ to the limit of infinitely large roots of $M(n)$, the single addends approach the cosine function,
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which again makes it easy to determine the value of the sum resp. integral, but
is it possible to prove by other means than Farey and Mertens, like for example trigonometric identities, that the real part is zero?
For the non-believers, here the simplest non-trivial example:
$$\scriptstyle{ \cos{6/23 \pi}+\cos{2/39 \pi}+\cos{14/39 \pi}-\cos{19/39 \pi}-\cos{5/39 \pi}-\cos{17/39 \pi}+\cos{2/21 \pi}+\cos{2/37 \pi}-\cos{13/37 \pi}-\cos{11/27 \pi}\\ +\cos{6/29 \pi}-\cos{17/37 \pi}+\cos{16/33 \pi}+\cos{10/27 \pi}+\cos{4/29 \pi}+\cos{4/25 \pi}-\cos{9/31 \pi}-\cos{11/37 \pi}-\cos{15/37 \pi}-\cos{7/33 \pi}\\-\cos{13/33 \pi}+\cos{8/39 \pi}+\cos{2/33 \pi}+\cos{8/37 \pi}-\cos{3/35 \pi}-\cos{9/23 \pi}+\cos{14/37 \pi}+\cos{8/23 \pi}+\cos{8/33 \pi}+\cos{2/35 \pi}\\-\cos{1/21 \pi}+\cos{8/21 \pi}+\cos{4/23 \pi}+\cos{14/33 \pi}-\cos{1/37 \pi}+\cos{4/31 \pi}-\cos{7/27 \pi}+\cos{4/37 \pi}-\cos{9/25 \pi}+\cos{8/25 \pi}\\-\cos{3/31 \pi}+\cos{10/31 \pi}+\cos{4/35 \pi}-\cos{7/23 \pi}+\cos{12/37 \pi}-\cos{13/35 \pi}+\cos{12/25 \pi}+\cos{12/35 \pi}+\cos{10/21 \pi}-\cos{9/37 \pi}+\cos{16/39 \pi}+\cos{10/37 \pi}-\cos{7/29 \pi}+\cos{2/29 \pi}-\cos{13/29 \pi}-\cos{11/23 \pi}-\cos{11/35 \pi}+\cos{10/33 \pi}-\cos{3/25 \pi}+\cos{10/23 \pi}+\cos{2/25 \pi}-\cos{5/27 \pi}+\cos{8/29 \pi}-\cos{3/29 \pi}+\cos{6/37 \pi}+\cos{6/31 \pi}+\cos{16/35 \pi}-\cos{3/37 \pi}-\cos{5/23 \pi}-\cos{7/31 \pi}\\-\cos{11/39 \pi}+\cos{2/23 \pi}-\cos{5/33 \pi}+\cos{8/27 \pi}-\cos{13/31 \pi}-\cos{1/27 \pi}+\cos{10/29 \pi}-\cos{1/35 \pi}-\cos{5/21 \pi}+\cos{10/39 \pi}\\-\cos{5/29 \pi}-\cos{1/33 \pi}+\cos{18/37 \pi}+\cos{12/29 \pi}+\cos{4/27 \pi}-\cos{1/31 \pi}+\cos{4/33 \pi}-\cos{7/25 \pi}+\cos{2/27 \pi}+\cos{2/31 \pi}\\+\cos{6/25 \pi}+\cos{14/29 \pi}-\cos{5/37 \pi}+\cos{14/31 \pi}-\cos{17/35 \pi}-\cos{1/29 \pi}+\cos{8/35 \pi}-\cos{13/27 \pi}-\cos{15/31 \pi}-\cos{1/23 \pi}\\+\cos{6/35 \pi}+\cos{4/39 \pi}-\cos{11/31 \pi}-\cos{11/29 \pi}+\cos{16/37 \pi}-\cos{11/25 \pi}-\cos{7/39 \pi}+\cos{12/31 \pi}+\cos{8/31 \pi}-\cos{9/29 \pi}\\+\cos{4/21 \pi}-\cos{7/37 \pi}-\cos{1/25 \pi}-\cos{5/31 \pi}-\cos{1/39 \pi}-\cos{9/35 \pi}-\cos{3/23 \pi} \overset{!}{=}0} $$
