Ramanujan continued fractions of order sixteen

We study the continued fractions [Formula: see text] and [Formula: see text] of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction [Formula: see text] with modularity and many interesting properties. Here we prove the modularities of [Formula: see text] and [Formula: see text] to find the relation with the generator of the field of modular functions on [Formula: see text]. Moreover we prove that the values [Formula: see text] and [Formula: see text] are algebraic integers for certain imaginary quadratic quantity [Formula: see text].

On Gauss sums over dedekind domains

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

Integral trace form of extensions of degree pq

In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text]. The integral trace form of [Formula: see text] allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of [Formula: see text].