I found this conjecture while working with the Möbius function

$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)} \, dx, $$

Apparently it seems to give correct results for formulae involving the Möbius functions (asymptotic) for example for the Riesz function $$ \sum_{n=0}^{\infty}\frac{\mu (n)}{n^{2}exp(-x/n^{2})} $$

But I can not give a serious complete proof of it :(