Explicit evaluation of certain sums of multiple zeta-star values

Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of \pi. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.Comment: 6 page

Hecke's integral formula for quadratic extensions of a number field

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the 0-th Laurent coefficients at s=1 of zeta functions of K and F.Comment: 16 page

On Kronecker limit formulas for real quadratic fields

Let $\zeta(s,C)$ be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to $\zeta'(0,C)$, when we change the signature of C.Comment: 24 page

On Shintani's ray class invariant for totally real number fields

We introduce a ray class invariant $X(C)$ for a totally real field, following Shintani's work in the real quadratic case. We prove a factorization formula $X=X_1... X_n$ where each $X_i=X_i(C)$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of $X_i(C)$ when the signature of $C$ at a real place is changed. This last result is also interpreted into an interesting behavior of the derivative $L'(0,\chi)$ of $L$-functions.Comment: 28 pages, 1 figur

Multiple zeta-star values and multiple integrals

We prove a kind of integral expressions for finite multiple harmonic sums and multiple zeta-star values. Moreover, we introduce a class of multiple integrals, associated with some combinatorial data (called 2-labeled posets). This class includes both multiple zeta and zeta-star values of Euler-Zagier type, and also several other types of multiple zeta values. We show that these integrals can be used to obtain some relations among such zeta values quite transparently

Multiple zeta functions of Kaneko-Tsumura type and their values at positive integers

Recently, a new kind of multiple zeta functions $\eta(k_1,...,k_r;s_1,...,s_r)$ was introduced by Kaneko and Tsumura. This is an analytic function of complex variables $s_1,...,s_r$, while $k_1,...,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $\eta(s'_1,...,s'_r;s_1,...,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove that there are some linear relations among these eta-values and the multiple zeta values $\zeta(k_1,...,k_r)$ of Euler-Zagier type.Comment: 14 page

Interpolation of multiple zeta and zeta-star values

We define polynomials of one variable t whose values at t=0 and 1 are the multiple zeta values and the multiple zeta-star values, respectively. We give an application to the two-one conjecture of Ohno-Zudilin, and also prove the cyclic sum formula for these polynomials.Comment: 13 page

On some multiple zeta-star values of one-two-three indices

In this paper, we present some identities for multiple zeta-star values with indices obtained by inserting 3 or 1 into the string 2,...,2. Our identities give analogues of Zagier's evaluation of \zeta(2,...,2,3,2,..., 2) and examples of a kind of duality of multiple zeta-star values. Moreover, their generalizations give partial solutions of conjectures proposed by Imatomi, Tanaka, Wakabayashi and the first author.Comment: 12 page

A new integral-series identity of multiple zeta values and regularizations

We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over Q. We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima's relation is discussed as well.Comment: 20 pages; Theorem 4.8 is added (v2); Some explanations (on eq.(2.4), Example 4.2, etc.) are added (v3

Shuffle product of finite multiple polylogarithms

In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zaiger and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition due to Komori-Matsumoto-Tsumura. As a corollary, we give an algebraic interpretation of our shuffle product.Comment: 13 page