Convergence of $-\sum_{n=1}^\infty\mu(n) J_0(2n)$, where $\mu(n)$ is the Möbius function and $J_0(y)$ the Bessel function of the first kind

Question

Let for integers $n\geq 1$ the Möbius function $\mu(n)$, see its definition from this MathWorld. And here $J_0(y)$ denotes the Bessel function of the first kind of order zero, see this MathWorld.

Question. What work can be done with the purpose to study if the series $$-\sum_{n=1}^\infty\mu(n) J_0(2n)$$ is convergent? Many thanks.

I don't know if our series is convergent. Using Wolfram Alpha online calculator one can to see that some partial sums, like next, of our series seem small in absolute value but erratic

sum mu(n)BesselJ(0,2n), from n=1 to 1000

sum mu(n)BesselJ(0,2n), from n=1 to 3000

Answer 1

Not a complete answer, but just a direction.

Using the large argument asymptotic for $J_0$ in here, we have $$ J_0(2n)=\sqrt{\frac1{\pi n}}\cos\left(2n-\frac{\pi}4\right)+O\left(\frac1{n\sqrt n}\right). $$ The sum of error term is absolutely convergent, so we now consider the convergence of $$ \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{\pi n}}\cos\left(2n-\frac{\pi}4\right). $$