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Toshiki Matsusaka

Publications and source records attributed to Toshiki Matsusaka.

At least 37 records · Page 2Linked to original sources
arXiv2024

A Hyperbolic Analogue of the Rademacher Symbol

One of the most famous results of Dedekind is the transformation law of $\log Δ(z)$. After a half-century, Rademacher modified Dedekind's result and introduced an $\mathrm{SL}_2(\mathbb{Z})$-conjugacy class invariant (integer-valued) function $Ψ(γ)$ called the Rademacher symbol. Inspired by Ghys' work on modular knots, Duke-Imamoglu-Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $Ψ_γ(σ)$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.

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arXiv2024

On integrality and asymptotic behavior of the $(k, l)$-Göbel sequences

Recently, Matsuhira, Matsusaka, and Tsuchida revisited old studies on the integrality of $k$-Göbel sequences and showed that the first 19 terms are always integers for any integer $k\ge 2$. In this article, we further explore two topics: Ibstedt's $(k,l)$-Göbel sequences and Zagier's asymptotic formula for the $2$-Göbel sequence, and extend their results.

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arXiv2023

Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients

Due to their singularities, multiple zeta functions behave sensitively at non-positive integer points. In this article, we focus on the asymptotic behavior at the origin $(0,\dots, 0)$ and unveil the generating series of the asymptotic coefficients as a generalization of the classical Gregory coefficients. This enables us to reveal the underlying symmetry of the asymptotic coefficients. Additionally, we extend the relationship between the asymptotic coefficients and the Gregory coefficients to include Hurwitz multiple zeta functions.

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arXiv2023

Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf-Witten invariant ${Z}_k$ for the ring of integers of the quadratic field $k=\mathbb{Q}(\sqrt{p_1\cdots p_r})$, where ${p_i}$'s are distinct prime numbers with $p_i \equiv 1 \pmod{4}$, and give a simple formula for $Z_k$ in terms of the graph obtained from quadratic residues among $p_1,\cdots, p_r$. Our result answers the question posed by Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf-Witten invariants.

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arXiv2023

Modular transformations of homological blocks for Seifert fibered homology $3$-spheres

In this article, for any Seifert fibered integral homology 3-sphere, we give explicit modular transformation formulas of homological blocks introduced by Gukov-Pei-Putrov-Vafa. Moreover, based on the modular transformation formulas, we have explicit asymptotic expansion formulas for the Witten-Reshetikhin-Turaev invariants which give a new proof of a version by Andersen of the Witten asymptotic conjecture.

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arXiv2023

How long can $k$-Göbel sequences remain integers?

Inspired by Episode 3 of the Japanese manga "Seisu-tan" by Doom Kobayashi and Shin-ichiro Seki, we investigate the $k$-Göbel sequence $(g_{k,n})_n$ named after Fritz Göbel. Although the sequence is generally defined as rational, quite a few initial terms behave like an integer sequence. This article addresses a question raised in Seisu-tan and shows that $g_{k,n}$ is always an integer for any $k \geq 2$ and $0 \leq n \leq 18$.

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arXiv2023

Curious congruences for cyclotomic polynomials II

We promote the recent research by Akiyama and Kaneko on the higher-order derivative values $Φ_n^{(k)}(1)$ of the cyclotomic polynomials. This article focuses on Lehmer's explicit formula of $Φ_n^{(k)}(1)/Φ_n(1)$ as a polynomial of the Euler and Jordan totient functions over $\mathbb{Q}$. Then we prove Akiyama-Kaneko's conjecture that the polynomials have a specific simple factor.

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arXiv2023

Hikami's observations on unified WRT invariants and false theta functions

The object of this article is a family of $q$-series originating from Habiro's work on the Witten-Reshetikhin-Turaev invariants. The $q$-series usually make sense only when $q$ is a root of unity, but for some instances, it also determines a holomorphic function on the open unit disc. Such an example is Habiro's unified WRT invariant $H(q)$ for the Poincaré homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of $H(ζ)$ at a root of unity is $1/2$ times the limit value of $H(q)$ as $q$ tends towards $ζ$ radially within the unit disc. In this article, we explain the appearance of the $1/2$-factor and generalize Hikami's observations by using Bailey's lemma and the theory of false theta functions.

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arXiv2022

Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=Γ_{2,3}$ by the triangle group $Γ_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\leq p<q$. We invoke the theory of harmonic Maass forms for $Γ_{p,q}$ to introduce the notion of the Rademacher symbol $ψ_{p,q}$, and provide several characterizations. Among other things, we generalize Ghys's theorem for modular knots around any "missing" torus knot $K_{p,q}$ in $S^3$ and in a lens space.

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arXiv2021

Bijective enumerations for symmetrized poly-Bernoulli polynomials

Recently, Bényi 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.

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arXiv2020

On the behavior of multiple zeta-functions with identical arguments on the real line

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $ζ_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical computations, and draw graphs of $ζ_r(s,s,\ldots,s)$ for real $s$, for several small values of $r$. Those graphs suggest various properties of $ζ_r(s,s,\ldots,s)$, some of which we prove rigorously. When $s \in [0,1]$, we show that $ζ_r(s,s,\ldots,s)$ has $r$ asymptotes at $\Re s=1/k$ ($1\leq k\leq r$), and determine the asymptotic behavior of $ζ_r(s,s,\ldots,s)$ close to those asymptotes. Numerical computations establish the existence of several real zeros for $2\leq r\leq 10$ (in which only the case $r=2$ was previously known). Based on those computations, we raise a conjecture on the number of zeros for general $r$, and gives a formula for calculating the number of zeros. We also consider the behavior of $ζ_r(s,s,\ldots,s)$ outside the interval $[0,1]$. We prove asymptotic formulas for $ζ_r(-k,-k,\ldots,-k)$, where $k$ takes odd positive integer values and tends to $+\infty$. Moreover, on the number of real zeros of $ζ_r(s,s,\ldots,s)$, we prove that there are exactly $(r-1)$ real zeros on the interrval $(-2n,-2(n-1))$ for any $n \geq 2$.

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arXiv2020

Linear Diophantine equations in Piatetski-Shapiro sequences

A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.

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arXiv2020

On the combinatorics of symmetrized poly-Bernoulli numbers

In this paper we introduce three combinatorial models for symmetrized poly-Bernoulli numbers. Based on our models we derive generalizations of some identities for poly-Bernoulli numbers. Finally, we set open questions and directions of further studies.

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arXiv2020

Symmetrized poly-Bernoulli numbers and combinatorics

Poly-Bernoulli numbers are one of generalizations of the classical Bernoulli numbers. Since a negative index poly-Bernoulli number is an integer, it is an interesting problem to study this number from combinatorial viewpoint. In this short article, we give a new combinatorial relation between symmetrized poly-Bernoulli numbers and the Dumont-Foata polynomials.

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arXiv2018

Traces of CM values and cycle integrals of polyharmonic Maass forms

Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the xi-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.

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arXiv2018

Polyharmonic weak Maass forms of higher depth for SL_2(Z)

The space of polyharmonic Maass forms was introduced by Lagarias-Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this paper, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct a basis as a generalization of Lagarias-Rhoades' works. As a corollary, we can obtain a preimage of an arbitrary polyharmonic weak Maass form under the xi-operator.

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