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In this article, using the arithmetic of Hurwitz zeta functions and Gauss sums, we investigate the arithmetic nature of ζ K (2 k + 1) ζ K + (2 k + 1) , where K is any totally imaginary abelian extension with maximal totally real subfield K + and k ≥ 1 is an integer. As a consequence, we derive an elementary proof of the elegant result of Murty and Pathak about irrationality of Dedekind zeta values associated to imaginary quadratic fields at odd integers. This alternate line of action allows us to link these questions to a question of Chowla and Milnor, resulting in some new results about Dedekind zeta values. [ABSTRACT FROM AUTHOR]

For any number field, we define Dedekind harmonic numbers with respect to this number field. First, we show that they are not integers except finitely many of them. Then, we present a uniform and an explicit version of this result for quadratic number fields. Moreover, by assuming the Riemann hypothesis for Dedekind zeta functions, we prove that the difference of two Dedekind harmonic numbers are not integers after a while if we have enough terms, and we prove the non-integrality of Dedekind harmonic numbers for quadratic number fields in another uniform way together with an asymptotic result. [ABSTRACT FROM AUTHOR]