Hardy-type investigation of Dedekind zeta zeros

Determine whether an analogue of Hardy's argument can be developed in the number-field setting to investigate non-trivial zeros of the Dedekind zeta function on the critical line, using the integral representation in the number-field extension of the Jacobi theta relation.

Background

The paper derives a number-field analogue of the Jacobi theta transformation and an associated integral representation involving the generalized Riemann Xi-function. For the rational number field, the corresponding classical integral representation was used by Hardy to prove that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line.

The authors explicitly ask whether the new number-field integral representation can be used to study non-trivial zeros of Dedekind zeta functions on the critical line, and more specifically whether Hardy's method can be adapted to the number-field setting. This question is not resolved in the paper, whose results establish the transformation and integral representations but do not derive a zero-distribution theorem for Dedekind zeta functions.