We define a combinatorial game in R from which we derive numerous new inequalities between higher-dimensional Dedekind sums. Our approach is motivated by a recent article by Dilcher and Girstmair, who gave a nice probabilistic interpretation for the classical Dedekind sum. Here we introduce a game analogous to Dilcher and Girstmair’s model in higher dimensions.

Using Weil’s explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.

The concepts of closure systems and closure operations in lattice theory are basic and applied to many fields in mathematics and theoretical computer science. In this paper authors find out a suitable definition of closure systems in Dedekind categories, and thereby give an equivalence proof for closure systems and closure operations in Dedekind categories.

Given a direct sum G of cyclic groups, we find a sharp bound for the minimal number of proper subgroups whose union is G. This problem generalizes to sums of cyclic modules over more general rings, such as local and Artinian rings or Dedekind domains and reduces to covering vector spaces by proper subspaces. As a consequence, we are also able to solve the analogous problem for monoids.

We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or ...

We investigate the modular group as a finitely presented group. It has a large collection of interesting quotients. In 1987 Conder substantially identified the onerelator quotients of the modular group which are defined using representatives of the 300 inequivalent extra relators with length up to 24. We study all such quotients where the extra relator has length up to 36. Up to equivalence, th...

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values.

A bipartite monoid is a commutative monoid Q together with an identified subset P ⊂ Q. In this paper we study a class of bipartite monoids, known as misère quotients, that are naturally associated to impartial combinatorial games. We introduce a structure theory for misère quotients with |P| = 2, and give a complete classification of all such quotients up to isomorphism. One consequence is that...