Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We describe the unit group of ${\Lambda }_{\wedge }\left[H\right]$ (the completion of the localization at $l$ of ${\mathbb{Z}}_l\left[\left[T\right]\right]\left[H\right]$) as well as the kernel and cokernel of the integral logarithm $L:{\Lambda }_{\wedge }{\left[H\right]}^{\times }\to {\Lambda }_{\wedge }\left[H\right]$, which appears in non-commutative Iwasawa theory.

1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_{\infty }$ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_{\infty }/k$. We know by the Ferrero-Washington...

Let $M\left(\alpha \right)$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $ℂ$. Then its Lehmer constant$L\left(V\right)$ is inf $M{\left(\alpha \right)}^{1/deg\left(\alpha \right)}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L\left(V\right)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L\left(V\right)$, which we call the transfinite Lehmer constant $L_{\infty }\left(V\right)$.Also, we prove the converse to Langevin’s Theorem, which states that $L\left(V\right)\>1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_{\infty }\left(V\right)$.

The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s $\Omega \left(3\right)$- conjecture for Galois extensions $K/F$ of number fields. It is certainly more difficult than the $\Omega \left(3\right)$-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that $F=\mathbb{Q}$ and the degree of $K/F$ is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important...

Let $\alpha$ be an algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\cdots ,{\alpha }_d$. In the paper we give a lower bound for the mean value$$M_p\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum }_{i=1}^d|log|{\alpha }_i{\left|\right|}^p}$$when $\alpha$ is not a root of unity and $p\>1$.

It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.