A note on a result of Bateman and Chowla
We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy's residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties...
The cross number κ(a) can be defined for any element a of a Krull monoid. The property κ(a) = 1 is important in the study of algebraic numbers with factorizations of distinct lengths. The arithmetic meaning of the weaker property, κ(a) ∈ ℤ, is still unknown, but it does define a semigroup which may be interesting in its own right. This paper studies some arithmetic(divisor theory) and analytic(distribution of elements with a given norm) properties of that semigroup and a related semigroup of ideals....
In this note we consider projective limits of Sinnott and Washington groups of circular units in the cyclotomic -extension of an abelian field. A concrete example is given to show that these two limits do not coincide in general.