We prove that every cyclic cubic extension $E$ of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in $E$. This extends the result of Schinzel who proved the same statement for every real quadratic field $E$. A corresponding conjecture is made for an arbitrary non-totally complex field $E$ and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.

1. Introduction et notations. Soit K un corps de nombres de degré n, de signature $(r_1,r_2)$ et de discriminant $d_K$. Dans [Od], A. M. Odlyzko évoque le problème de savoir l’ordre de grandeur du premier zéro de la fonction zêta de Dedekind. Dans cette direction, une conjecture a été énoncée dans [To] qui dit que la hauteur du premier zéro est majorée par $C/ln\left(\right|d_K\left|\right)$ où C est une constante positive qui ne dépend que de n. L’idée de cette dernière inégalité provient d’un théorème de densité (sous GRH) dû a S. Lang [La1]....

Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal ${}^{\ell }$. We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $Gal(K\left(\sqrt[\ell ]{a_{}}\right)/K)$ for every choice of $a_{}$. We then show that becomes principal in L if and only if every reciprocal prime is not a norm inside...

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...

In the previous paper [15], we determined the structure of the Galois groups $\text{Gal}(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $\geqq 723$) and give a table of $\text{Gal}(K_{ur}/K)$. We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $\text{Gal}(K_{ur}/K)$. In particular, for $K=𝐐\left(\sqrt{-856}\right)$, we obtain $\text{Gal}(K_{ur}/K)\cong \widetilde{S_4}\times C_5\text{and}{K}_{ur}=K_4$, the fourth Hilbert class field of $K$. This is the first example of a number field whose...

We determine the structures of the Galois groups Gal$(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 420(\leqq 719$ under the Generalized Riemann Hypothesis). For all such $K$, $K_{ur}$ is $K$, the Hilbert class field of $K$, the second Hilbert class field of $K$, or the third Hilbert class field of $K$. The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We also use class...

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$S_{l,K_3}\left(x\right)={\sum }_{m⩽x}{M}^l\left(m\right)$$ , where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$S_{2,K_3}\left(x\right)$$ and $$S_{3,K_3}\left(x\right)$$ .