Infinite Product of the Riemann auxiliary function

We obtain the product for the auxiliary function $\mathop{\mathcal R}(s)$ and study some related functions as its phase $\omega(t)$ at the critical line. The function $\omega(t)$ determines the zeros of $\zeta(s)$ on the critical line. We study the influence of the zeros of $\mathop{\mathcal R}(s)$ on $\omega(t)$. Thus, the relationship between the zeros of $\mathop{\mathcal R}(s)$ and those of $\zeta(s)$ is determined.