Traces of certain integral operators related to the Riemann hypothesis

We prove the existence of a nontrivial singular trace $\tau$ defined on an ideal $\mathcal{J}$ closed with respect to the logarithmic submajorization such that $\tau (A_{\rho }(\alpha ))=0$, where $A_{\rho }(\alpha ):L^2(0,1)\to L^2(0,1)$, $[A_{\rho }(\alpha )f](\theta )={\int }_0^1\rho (\alpha \theta /x)f(x)dx$, $0<\alpha \le 1$. We also show that $\tau (A_{\rho }(\alpha ))=0$ for every $\tau$ nontrivial singular trace on $\mathcal{J}$. Finally, we give a recursion formula from which we can evaluate all the traces $\text{Tr}(A_{\rho }^r(\alpha ))$, $r\in \mathbb{N}$, $r\ge 2$.