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For any cubic number field K and any conductor ideal f of K we describe how to find all orders of K with conductor f . The result depends only on the factorization of the rational prime numbers in K , and it also yields an elementary proof for a formula for the zeta function of orders in a cubic number field. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the Ohno–Zagier relation of multiple zeta values can be deduced from the regularized double shuffle relation. At the same time, we provide an algebraic proof of Ohno–Zagier relation. [Copyright &y& Elsevier]

Abstract: Let K be a real quadratic field with discriminant d, and for a (fractional) ideal a of K, let Na be the norm of a. For a given fractional ideal I of K, and Dirichlet character χ of conductor q, we define where the sum is over all integral ideals a of K which are equivalent to I. We give a short, easily computable formula to evaluate , using familiar objects from considerations of K. We generalize our formula to with , though the result obtained is not quite so satisfactory as that for . We discuss connections between these formulae and small class numbers. [Copyright &y& Elsevier]

Abstract: In this paper, we are interested in the average behavior of the coefficients of Dedekind zeta function over square numbers. In Galois fields of degree d which is odd, when is an integer, we have where , is a polynomial in t of degree , and is an arbitrarily small constant. By using our method, we also rectify the main terms of the k-dimensional divisor problem in some Galois fields over square numbers established by Deza and Varukhina (2008) . [Copyright &y& Elsevier]

Abstract: We introduce two types of zeta functions (Ψ-type and ζ-type) of one complex variable associated to an arbitrary number field K. We prove various arithmetic identities which involve both of them. We also study their special values at integral arguments. [Copyright &y& Elsevier]

Abstract: In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng''s rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable -lattices with rank r. For , in the case of , Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng''s rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions. [Copyright &y& Elsevier]

Abstract: In this paper, we show the following theorems. Suppose are algebraically independent numbers and for . Then we have the joint t-universality for Lerch zeta functions for . Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions. [Copyright &y& Elsevier]