Zeta Functions Over Zeros of General Zeta and L-Functions [chapter]

André Voros
Zeta Functions, Topology and Quantum Physics  
We describe in detail three distinct families of generalized zeta functions built over the nontrivial zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only. Explicit properties are also displayed more clearly than before. Several tables of formulae cover the simplest concrete cases: L-functions for real primitive Dirichlet characters, and Dedekind zeta functions. Generalities This text is a partial expansion of our
more » ... l expansion of our oral presentation, which surveyed an earlier paper [27] on zeta functions over the Riemann zeros: these were Dirichlet series built out of the nontrivial zeros of a "primary" zeta function, Riemann's ζ(x), thus defining newer or "secondary" [5] zeta functions. Here we will fully develop the argument of [27, Sec. 5.5], which indicated how the primary function can actually be taken more general than just ζ(x); we also incorporate and extend subsequent work [28] . Accordingly, we can now
doi:10.1007/0-387-24981-8_10 fatcat:jf5sz42vfrgxpn5szdmoios3km