The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Mani...

It is known that the 3-manifold SL(2,Z)\ SL(2,R) is diffeomorhic to the complement of the trefoil knot in S3. As is shown by E. Ghys the linking number of the trefoil with a modular knot associated to a hyperbolic conjugacy is related to the classical Dedekind symbol. These symbols arose historically in the transformation property of the logarithm of Dedekind’s eta function. In this paper we st...

We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg tra...

Abstract: In this paper, with the help of the Hardy and Dedekind sums we will give many properties of the sum B1(h, k), which was defined by Cetin et al. Then we will give the connections of this sum with the other well-known finite sums such as the Dedekind sums, the Hardy sums, the Simsek sums Y(h, k) and the sum C1(h, k). By using the Fibonacci numbers and two-term polynomial relation, we wi...

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers a and b giving

We give a definition of class Dedekind domains which includes the rings integers global fields and proof that all in this have finite ideal group. also prove coincides with fields.

We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L-functions).

We study the asymptotic behavior of the classical Dedekind sums s(sk/tk) for the sequence of convergents sk/tk k ≥ 0, of the transcendental number

The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction...

We investigate homomorphisms of degree structures with various relations, functions and constants. Our main emphasis is on pseudolattices, i.e., partially ordered sets with a join operation and relations simulating the meet operation. We show that there are no 5nite quotients of the pseudolattice of degrees or of the pseudolattice of degrees 6 0′, but that many 5nite distributive lattices are p...