Zeta functions for Simplicial Complexes
For a finite abstract simplicial complex, we can look at the Dirac Zeta function, the Connection Zeta function and the Lefschetz zeta function. My work on this in the last couple of years
Tag: zeta function
A Dyadic Riemann hypothesis
When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.
Hearing the shape of a simplicial complex
A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?