On higher order real zeros of Dedekind zeta functions

Abstract In 1972, Heilbronn introduced the notion of virtual characters and used it to study simple real zeros of Dedekind zeta functions. One of the consequences of his elegant work is the following. Let $\mathrm{K}/ \mathrm{F}$ be a Galois extension of number fields of odd degree. Then any real simple zero of $\zeta_{\mathrm{K}}$ is necessarily a simple zero of $\zeta_{\mathrm{F}}$ . The ethos of this paper is to carry out investigation for arbitrary odd order real zeros. While the Riemann zeta function is conjectured to have only simple zeros, the same does not hold for arbitrary Dedekind zeta functions. One of the consequences of our work is that any Galois number field K of odd degree cannot have a non-trivial odd order real zero. Such parity is at least in conformity with extended Riemann hypothesis as the order of vanishing of the Dedekind zeta function $\zeta_{\mathrm{K}}$ at $1/2$ is necessarily even. We also indicate, via a number of illustrative examples (see Remarks 1·1 and 1·2), that in some sense our results are optimal.