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Search: MSC category 11R42 ( Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] )

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1. CMB Online first

Sands, Jonathan W.
$L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Let $n$ be a positive even integer, and let $F$ be a totally real number field and $L$ be an abelian Galois extension which is totally real or CM. Fix a finite set $S$ of primes of $F$ containing the infinite primes and all those which ramify in $L$, and let $S_L$ denote the primes of $L$ lying above those in $S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$-integers of $L$. Suppose that $\psi$ is a quadratic character of the Galois group of $L$ over $F$. Under the assumption of the motivic Lichtenbaum conjecture, we obtain a non-trivial annihilator of the motivic cohomology group $H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the $S$-modified Artin $L$-function $L_{L/F}^S(s,\psi)$ at $s=1-n$.

Keywords:motivic cohomology, regulator, Artin L-functions
Categories:11R42, 11R70, 14F42, 19F27

2. CMB Online first

Mantilla-Soler, Guillermo
Weak arithmetic equivalence
Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses, this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence.

Keywords:arithmeticaly equivalent number fields, root numbers
Categories:11R04, 11R42

3. CMB 2009 (vol 52 pp. 186)

Broughan, Kevin A.
Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma + {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is shown to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$ the inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small and to the region $\si\ge 1+1/(\log t -5)$ for all $t\neq 0$. This answers positively a question posed by Lagarias.

Keywords:Riemann zeta function, xi function, zeta zeros
Categories:11M26, 11R42

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