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Under GRH, we establish a version of Duke's short-sum theorem for entire Artin L -functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit. [ABSTRACT FROM AUTHOR]

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for x ≥ 2 . Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at s = 1 . [ABSTRACT FROM AUTHOR]

We give an extension of a theorem of Cesàro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai's function to number fields. [ABSTRACT FROM AUTHOR]

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ , i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field. [ABSTRACT FROM AUTHOR]

In 2008, Deza and Varukhina established asymptotic formula for the mean value of the arithmetic function $$\tau _{k_1}^K(n)\tau _{k_2}^K(n)\cdots \tau _{k_l}^K(n)$$ , where $$K$$ is a quadratic or cyclotomic field, and $$\tau _{k}^K(n)$$ is the $$k$$ -dimensional divisor function in the number field $$K$$ . Recently, Lü generalized their results to any Galois extension $$K$$ of the rational field. It seems interesting to deal with similar problems which involve different number fields. In this paper, we are concerned with the mean value of the arithmetic function $$\tau _{k_1}^{K_1}(n)\tau _{k_2}^{K_2}(n)\cdots \tau _{k_l}^{K_l}(n)$$ , where $$K_j$$ are number fields whose discriminants are relatively prime. [ABSTRACT FROM AUTHOR]

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/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M( m) denotes the number of integral ideals of the field K of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$. [ABSTRACT FROM AUTHOR]

Deza and Varukhina [3] established asymptotic formulae for some arithmetic functions in quadratic and cyclotomic fields. We generalize their results to any Galois extension of the rational field. During this process we rectify the main terms in their asymptotic formulae. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. Assume that ζK(s)/ζ(s) is entire. We give an explicit upper bound for the residue at s = 1 of the Dedekind zeta function ζK(s) of K. We deduce explicit upper bounds on class numbers of cubic and quartic number fields. [ABSTRACT FROM AUTHOR]