Traces of certain integral operators related to the Riemann hypothesis

Abstract

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^{1}_{0}\rho(\alpha\theta/x)f(x)dx} $, $ 0 < \alpha\leq 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 $ {\mbox{Tr}}\, (A^{r}_{\rho}(\alpha)) $, $ r\in \mathbb{N} $, $ r\geq 2 $.

Keywords

• singular trace
• spectral trace
• modified fredholm determinant
• riemann hypothesis