In 1988, Gross proposed a conjectural congruence between Stickelberger
elements and algebraic regulators, which is often referred to as the refined
class number formula. In this paper, we prove this congruence.

arXiv

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Other Link： http://arxiv.org/pdf/1608.04718v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s00229-020-01191-5

arXiv

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Other Link： http://link.springer.com/article/10.1007/s00229-020-01191-5/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s11139-020-00341-3

arXiv

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Other Link： http://link.springer.com/article/10.1007/s11139-020-00341-3/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：Faculty of Mathematics, Kyushu University  

DOI： 10.2206/kyushujm.75.115

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：International Press of Boston  

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.

DOI： 10.4310/ajm.2021.v25.n6.a4

arXiv

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Other Link： http://arxiv.org/pdf/1905.04875v1

Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jalgebra.2020.01.032

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.indag.2020.04.004

arXiv

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In this paper, we introduce a new connector which generalizes the connector
found by the third author and Yamamoto. The new connector gives a direct proof
of the double Ohno relation recently proved by the first author, the second
author, Murahara, and Onozuka. Furthermore, we obtain a simultaneous
generalization of the ($q$-)Ohno relation and the ($q$-)double Ohno relation.

arXiv

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Other Link： http://arxiv.org/pdf/2006.09036v1

Publishing type：Research paper (scientific journal)   Publisher：Aarhus University Library  

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

DOI： 10.7146/math.scand.a-119209

arXiv

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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 ${\rm SL}_{2}(\mathbb{Z})$.

arXiv

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Other Link： http://arxiv.org/pdf/2003.05236v1

Publishing type：Research paper (scientific journal)   Publisher：European Mathematical Society Publishing House  

DOI： 10.4171/prims/56-1-9

arXiv

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Symmetric multiple zeta values (SMZVs) are elements in the ring of all
multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by
Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that
symmetric multiple zeta values satisfy double shuffle relations and duality
relations. In this paper, we construct certain lifts of SMZVs which live in the
ring generated by all multiple zeta values and $2\pi i$ as certain iterated
integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed
path. We call this lifted values as refined symmetric multiple zeta values
(RSMZVs). We show double shuffle relations and duality relations for RSMZVs.
These relations are refinements of the double shuffle relations and the duality
relations of SMZVs. Furthermore we compare RSMZVs to other variants of lifts of
SMZVs. Especially, we prove that RSMZVs coincide with
Bachmann-Takeyama-Tasaka's $\xi$-values.

arXiv

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Other Link： http://arxiv.org/pdf/1807.04747v2

We consider a cyclic analogue of multiple zeta values (CMZVs), which has two
kinds of expressions; series and integral expression. We prove an
`integral$=$series' type identity for CMZVs. By using this identity, we
construct two classes of $\mathbb{Q}$-linear relations among CMZVs. One of them
is a generalization of the cyclic sum formula for multiple zeta-star values. We
also give an alternative proof of the derivation relation for multiple zeta
values.

DOI： 10.14992/00018670

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jnt.2018.11.023

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.aim.2019.03.005

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jnt.2018.05.019

Web of Science

Scopus

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on [Formula: see text] for our proof.

DOI： 10.1142/s1793042119500052

Web of Science

arXiv

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Language：English  

In this paper we prove a weighted sum formula for multiple harmonic sums
modulo primes, thereby proving a weighted sum formula for finite multiple zeta
values. Our proof utilizes difference equations for the generating series of
multiple harmonic sums. We also conjecture several weighted sum formulas of
similar flavor for finite multiple zeta values.

DOI： 10.1090/proc/14588

arXiv

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Other Link： http://arxiv.org/pdf/1808.00844v1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

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.

DOI： 10.1007/s00209-015-1467-y

Web of Science

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Language：English   Publisher：SPRINGER  

From the theory of modular forms, there are exactly $[(k-2)/6]$ linear
relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\
(2\le i \le [k/4])$. We present explicit formulas among these modular forms
based on the partial fraction decomposition, and use them to determining a
basis of the space of modular forms of weight $k$ on ${\rm SL}_2({\mathbb Z})$.

DOI： 10.1007/s11139-014-9639-7

Web of Science

arXiv

J-GLOBAL

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Other Link： http://arxiv.org/pdf/1402.1585v1

Language：English   Publishing type：Research paper (scientific journal)  

The purpose of this paper is to prove a conjecture in Yoshida's book [2, p.33] on the higher derivative of Shintani zeta functions at s = 0. We use multivariable Shintani zeta functions to prove the conjecture. © 2011 The Japan Academy.

DOI： 10.3792/pjaa.87.13

Scopus

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We consider the symmetric multiple zeta values in $\mathcal{S}_m$ without
modulo $\pi^2$ reduction for indices in which $1$ and $3$ appear alternately.
We investigate those values that can be expressed as a polynomial of the
Riemann zeta values, and give a conjecturally complete list of explicit
formulas for such values.

arXiv

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Other Link： http://arxiv.org/pdf/2203.07701v1

Parametrized multiple series are generalizations of the multiple zeta values
introduced by Igarashi. In this work, we completely determine all the linear
relations among these parameterized multiple series. Specifically, we prove the
following two statements: the linear part of the Kawashima relation for
multiple zeta values can be generalized to the parametrized multiple series;
any linear relations among the parametrized multiple series can be written as a
linear combination of the linear part of the Kawashima relation.

arXiv

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Other Link： http://arxiv.org/pdf/2111.12975v1

Recently, B\'{e}nyi and the second author introduced two combinatorial
interpretations for symmetrized poly-Bernoulli polynomials. In the present
study, we construct bijections between these combinatorial objects. We also
define various combinatorial polynomials and prove that all of these
polynomials coincide with symmetrized poly-Bernoulli polynomials.

arXiv

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Other Link： http://arxiv.org/pdf/2107.11952v1

The Ohno relation for multiple zeta values can be formulated as saying that a
certain operator, defined for indices, is invariant under taking duals. In this
paper, we generalize the Ohno relation to regularized multiple zeta values by
showing that, although the suitably generalized operator is not invariant under
taking duals, the relation between its values at an index and at its dual index
can be written explicitly in terms of the gamma function.

arXiv

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Other Link： http://arxiv.org/pdf/2105.09631v1

We prove the Ohno-type relation for the interpolated multiple zeta values,
which was introduced first by Yamamoto. Same type results for finite multiple
zeta values are also given. Moreover, these relations give the sum formula for
interpolated multiple zeta values and interpolated $\mathcal{F}$-multiple zeta
values, which were proved by Yamamoto and Seki, respectively.

arXiv

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Other Link： http://arxiv.org/pdf/2103.14851v1

In this paper we introduce confluence relations for motivic Euler sums (also
called alternating multiple zeta values) and show that all linear relations
among motivic Euler sums are exhausted by our confluence relations. This
determines all automorphisms of the de Rham fundamental torsor of
$\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the
motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we
also discuss other applications of our confluence relations such as an explicit
$\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis and
$2$-adic integrality of the coefficients in the expansion.

arXiv

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Other Link： http://arxiv.org/pdf/2007.04288v1

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 ${\rm SL}_{2}(\mathbb{Z})$.

arXiv

researchmap

Other Link： http://arxiv.org/pdf/2003.05236v1

Publishing type：Internal/External technical report, pre-print, etc.  

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.

arXiv

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Other Link： http://arxiv.org/pdf/1808.01559v1

Publishing type：Internal/External technical report, pre-print, etc.  

In 1988, Gross proposed a conjectural congruence between Stickelberger
elements and algebraic regulators, which is often referred to as the refined
class number formula. In this paper, we prove this congruence.

arXiv

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Other Link： http://arxiv.org/pdf/1608.04718v1

Publishing type：Internal/External technical report, pre-print, etc.  

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.

arXiv

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Other Link： http://arxiv.org/pdf/1602.00666v1

Language：Japanese   Publisher：京都大学  

CiNii Books

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Publishing type：Internal/External technical report, pre-print, etc.  

We define the class of normalized Shintani L-functions of several variables.
Unlike Shintani zeta functions, the normalized Shintani L-function is a
holomorphic function. Moreover it satisfies a good functional equation. We show
that any Hecke L-function of a totally real field can be expressed as a
diagonal part of some normalized Shintani L-function of several variables. This
gives a good several variables generalization of a Hecke L-function of a
totally real field. This also gives a new proof of the functional equation of
the Hecke L-function of a totally real field.

arXiv

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Other Link： http://arxiv.org/pdf/1312.6218v1

Language：Japanese   Publisher：京都大学  

CiNii Books

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Venue：Kyoto University  

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Venue：Kyushu University  

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Venue：Hokkaido University  

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Presentation type：Oral presentation (general)  

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Presentation type：Oral presentation (general)  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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