Order zeta functions of number rings and resolution of singularities

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The project aims at the study of fundamental arithmetic and analytic invariants of arithmetically motivated zeta functions such as order zeta functions of number rings. The latter are Dirichlet-type generating series enumerating order (subrings with one) of rings of integers in algebraic number fields.In contrast to the classical theory of the related Dedekind zeta function, the fundamental analytic invariants of these functions -- such as their abscissae of convergence, pole orders, special values etc -- are largely unknown. A conjecture attributed to Bhargava has implications on the abscissa of convergence of order zeta functions and hence on the degree of polynomial growth of orders in number rings. Newer papers by Kaplan e.a. yield some estimates for the invariants we mentioned. Explicit formulae for order zeta functions, however, are only known for number fields of degree less than five.The zeta functions studied in the project all have natural Euler product decompositions, whose factors are rational functions. The known formulae for number fields of small degree suggest a number of deep arithmetic regularity, uniformity, and symmetry phenomena. Their detailed study lies at the heart of the proposal.An established method to study the relevant Euler factors and their products interprets the factors as suitable p-adic integrals. A uniform understanding of these integrals therefore holds the key to the understanding both of the global zeta functions and the arithmetic properties of their Euler factors. Resolution of singularities of associated hypersurfaces are a central tool in this enterprise. Whilst Hironaka's celebrated theorem guarantees the existence of such resolutions, the existing algorithms usually soon yield to the complexity and high-dimensionality of the relevant hypersurfaces. A key idea of the proposed project is to use the symmetries and recursive structures occurring in the specific arithmetic context of order zeta functions of number rings in order to design and implement taylor-made resolutions of singularities in this context.The project brings together an experienced researcher and practitioner in the field of resolutions of singularities and an expert in the field of zeta functions of groups and rings. The combination of the two PIs' respective expertise promises significant progress in a field of asymptotic ring theory of high current international relevance.