Spectral action with zeta function regularization

In this paper we propose a novel definition of the bosonic spectral action using zeta function regularization, in order to address the issues of renormalizability and spectral dimensions. We compare the zeta spectral action with the usual (cutoff based) spectral action and discuss its origin, predictive power, stressing the importance of the issue of the three dimensionful fundamental constants, namely the cosmological constant, the Higgs vacuum expectation value, and the gravitational constant. We emphasize the fundamental role of the neutrino Majorana mass term for the structure of the bosonic action.