Shintani zeta functions and a refinement of Gross's leading term conjecture

We introduce the notion of Shintani data, which axiomatizes algebraic aspects of Shintani zeta functions. We develop the general theory of Shintani data, and show that the order of vanishing part of Gross's conjecture follows from the existence of a Shintani datum. Recently, Dasgupta and Spiess proved the order of vanishing part of Gross's conjecture under certain conditions. We give an alternative proof of their result by constructing a certain Shintani datum. We also propose a refinement of Gross's leading term conjecture by using the theory of Shintani data. Out conjecture gives a conjectural construction of localized Rubin-Stark elements which can be regarded as a higher rank generalization of the conjectural construction of Gross-Stark units due to Dasgupta and Dasgupta-Spiess.Comment: 23 page

On the functional equation of the normalized Shintani l-function of several variables

In this paper, we introduce the normalized Shintani L-function of several variables by an integral representation and prove its functional equation. The Shintani L-function is a generalization to several variables of the Hurwitz-Lerch zeta function and the functional equation given in this paper is a generalization of the functional equation of Hurwitz-Lerch zeta function. In addition to the functional equation, we give special values of the normalized Shintani L-function at non-positive integers and some positive integers

On Hoffman's conjectural identity

In this paper, we shall prove the equality $$\zeta(3,\{2\}^{n},1,2)=\zeta(\{2\}^{n+3})+2\zeta(3,3,\{2\}^{n})$$ conjectured by Hoffman using certain identities among iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$.Comment: 4 page

Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values

Ohno's relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama's theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three

Sum formula for multiple zeta function

The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function

Polynomial generalization of the regularization theorem for multiple zeta values

Ihara, Kaneko, and Zagier defined two regularizations of multiple zeta values and proved the regularization theorem that describes the relation between those regularizations. We show that the regularization theorem can be generalized to polynomials whose coefficients are regularizations of multiple zeta values and that specialize to symmetric multiple zeta values defined by Kaneko and Zagier.Comment: 7 pages; typos correcte

An interpolation of Ohno's relation to complex functions

Ohno's relation is a well known formula among multiple zeta values. In this paper, we present its interpolation to complex functions

Modular phenomena for regularized double zeta values

In this paper, we investigate linear relations among regularized motivic iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ of depth two, which we call regularized motivic double zeta values. Some mysterious connection between motivic multiple zeta values and modular forms are known, e.g. Gangl-Kaneko-Zagier relation for the totally odd double zeta values and Ihara--Takao relation for the depth graded motivic Lie algebra. In this paper, we investigate so-called non-admissible cases and give many new Gangl-Kaneko-Zagier type and Ihara-Takao type relations for regularized motivic double zeta values. Especially, we construct linear relations among a certain family of regularized motivic double zeta values from odd period polynomials of modular forms for the unique index two congruence subgroup of the full modular group. This gives the first non trivial example of a construction of the relations among multiple zeta values (or their analogues) from modular forms for a congruence subgroup other than the ${\rm SL}_{2}(\mathbb{Z})$.Comment: 22 pages, comments welcome

Ohno type relations for classical and finite multiple zeta-star values

Ohno's relation is a generalization of both the sum formula and the duality formula for multiple zeta values. Oyama gave a similar relation for finite multiple zeta values, defined by Kaneko and Zagier. In this paper, we prove relations of similar nature for both multiple zeta-star values and finite multiple zeta-star values. Our proof for multiple zeta-star values uses the linear part of Kawashima's relation.Comment: 9 page

Duality/Sum formulas for iterated integrals and their application to multiple zeta values

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given.Comment: 11 page