Searcharxiv⌕ Search

arXiv subjects

Shingo Saito

Publications and source records attributed to Shingo Saito.

At least 19 recordsLinked to original sources
arXiv2024

Ohno relation for regularized refined symmetric multiple zeta values

The Ohno relation is one of the most celebrated results in the theory of multiple zeta values, which are iterated integrals from $0$ to $1$. In a previous paper, the authors generalized the Ohno relation to regularized multiple zeta values, which are non-admissible iterated integrals from $0$ to $1$. Meanwhile, Takeyama proved an analogue of the Ohno relation for refined symmetric multiple zeta values, which are iterated integrals from $0$ to $0$. In this paper, we generalize Takeyama's result to regularized refined symmetric multiple zeta values, which are non-admissible iterated integrals from $0$ to $0$.

math.NT↗
arXiv2024

Characterization of order structures avoiding three-term arithmetic progressions

It is known that the set of all nonnegative integers may be equipped with a total order that is chaotic in the sense that there is no monotone three-term arithmetic progressions. Such chaotic order must be so complicated that the resulting ordered set cannot be order isomorphic to the set of all nonnegative integers or the set of all integers with the standard order. In this paper, we completely characterize order structures of chaotic orders on the set of all nonnegative integers, as well as on the set of all integers and on the set of all rational numbers.

math.CO↗
arXiv2022

Interpolated polynomial multiple zeta values of fixed weight, depth, and height

We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.

math.NT↗
arXiv2021

Ohno relation for regularized multiple zeta values

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.

math.NT↗
arXiv2020

Generating functions for sums of polynomial multiple zeta values

The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.

math.NT↗
arXiv2020

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.

math.NT↗
arXiv2019

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.

math.NT↗
arXiv2018

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.

math.NT↗
arXiv2018

Weighted sum formula for multiple harmonic sums modulo primes

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.

math.NT↗
arXiv2018

Restricted sum formula for finite and symmetric multiple zeta values

The sum formula for finite and symmetric multiple zeta values, established by Wakabayashi and the authors, implies that if the weight and depth are fixed and the specified component is required to be more than one, then the values sum up to a rational multiple of the analogue of the Riemann zeta value. We prove that the result remains true if we further demand that the component should be more than two or that another component should also be more than one.

math.NT↗
arXiv2014

Bowman-Bradley type theorem for finite multiple zeta values

The multiple zeta values are multivariate generalizations of the values of the Riemann zeta function at positive integers. The Bowman-Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between 3,1,...,3,1 add up to a rational multiple of a power of π. We show that an analogous theorem holds in a very strong sense for finite multiple zeta values, which have been investigated by Hoffman and Zhao among others and recently recast by Zagier.

math.NT↗
arXiv2013

Sum formula for finite multiple zeta values

The sum formula is one of the most well-known relations among multiple zeta values. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.

math.NT↗
arXiv2012

Knot points of typical continuous functions

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.

math.CA↗
arXiv2010

Combinatorial remarks on the cyclic sum formula for multiple zeta values

The multiple zeta values are generalizations of the values of the Riemann zeta function at positive integers. They are known to satisfy a number of relations, among which are the cyclic sum formula. The cyclic sum formula can be stratified via linear operators defined by the second and third authors. We give the number of relations belonging to each stratum by combinatorial arguments.

math.NT↗
arXiv2010

The Bowman-Bradley theorem for multiple zeta-star values

The Bowman-Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between 3,1,...,3,1 add up to a rational multiple of a power of pi. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.

math.NT↗
arXiv2008

Continuity Points of Typical Bounded Functions

Kostyrko and Salat showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples.

math.CA↗