Cesàro–Fejér Regularization of Dirichlet Series and Spectral Detection of Riemann Zêta Zeros

We show that Riemann Zêta function admits a regularization, which leads to an equivalent integral form through the Fejér Kernel. Upon a particular window resetting we obtain a representation through a sinc² which we bridge to the Dyson-Montgomery RMT GUE sine kernel as a rigid semi classical counterpart. The multiplicative structure allows to find Euler prime formula factorisation, while the regularization additively "thermalizes" the Dirichlet serie representation of the RZ function. Then this function can be seen as a Mellin-Fourier projection of a regular additive measure on the multiplicative support, which action is minimized around the non trivial zeros. Thus exhibiting a "minimal entropy" configuration of the measure.