Toshiki Matsusaka

Modular integrals for individual modular knots

Toshiki Matsusaka

To each modular knot, we attach a weight $2$ modular integral for $SL_{2} (Z)$ whose homogenized cycle integrals recover its linking numbers with other modular knots…

Distribution of Kaneko's val function

Toshiki Matsusaka

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$ -function along closed geodesics on the modular surface. We prove that, when primitive…

Some results on naive transcendence in the ring of integers modulo infinitely large primes

Toshiki Matsusaka, Shin-ichiro Seki

This paper presents various transcendence results in the ring of integers modulo infinitely large primes $A$ . In the ring $A$ , one can consider two notions of t…

On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials

Toshiki Matsusaka, Taichi Miyazaki, Shunta Yara

We study a finite analogue of Dobiński's formula, which is related to the Napier constant $e$ , and its Bessel-type generalizations. Furthermore, using Gregory polynomials, we exten…

Poly-Bernoulli numbers from shifted log-sine integrals

Toshiki Matsusaka

In 1999, Arakawa and Kaneko introduced a zeta function connecting multiple zeta values with poly-Bernoulli numbers through its special values. Log-sine integrals are also known to…

math.GT2025

On the Alexander polynomials of modular knots

Soon-Yi Kang, Toshiki Matsusaka, Kyungbae Park

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $SL_{2} (Z)$ …

The Fourier coefficients and singular moduli of the elliptic modular function $j (τ)$ , revisited

Toshiki Matsusaka

Kaneko's formula expresses the Fourier coefficients of the elliptic modular $j$ -function as finite sums of singular moduli. First published as a short article in 1996, it was prese…

Applications of Faà di Bruno's formula to partition traces

Toshiki Matsusaka

We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of pa…

Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers

Toshiki Matsusaka, Miyu Suzuki

We extend the recently developed theory of Roehrig and Zwegers on indefinite theta functions to prove certain power series are modular forms. As a consequence, we obtain several po…

Hecke equivariance of the divisor map

Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim +1

We study the multiplicative Hecke operators acting on the space of meromorphic modular forms, and show that the divisor map to divisors on $X_{0} (N)$ is a Hecke equivariant map. As a…

Denominator identity for the affine Lie superalgebra $spo (2 m, 2 m + 1)$ and indefinite theta functions

Toshiki Matsusaka, Miyu Suzuki

In 1994, Kac and Wakimoto found the denominator identity for classical affine Lie superalgebras, generalizing that for affine Lie algebras. As an application, they obtained power s…

Quasi-modularity in MacMahon partition variants and prime detection

Soon-Yi Kang, Toshiki Matsusaka, Gyucheol Shin

Building on the results of Craig, van Ittersum, and Ono, we provide a refined understanding of MacMahon's partition functions and their variants, including their quasi-modular prop…

A unified approach to Rohrlich-type divisor sums

Daeyeol Jeon, Soon-Yi Kang, Chang Heon Kim +1

We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $Γ_{0} (N)$ . Our main theorem unifies various results from the literature, a…

Hauptmoduln and even-order mock theta functions modulo 2

Soon-Yi Kang, Seonkyung Kim, Toshiki Matsusaka +1

The Fourier coefficients $c_{1} (n)$ of the elliptic modular $j$ -function are always even for $n \neq \equiv 7 (mod 8)$ . In contrast, for $n \equiv 7 (mod 8)$ , it is conjectured that…

Note on an explicit formula of Rademacher symbols for triangle groups

Toshiki Matsusaka, Gyucheol Shin

The purpose of this note is to present an explicit formula of the Rademacher symbols for triangle groups. This result generalizes Ghys' third proof of the identity relating to the…

Eichler-Selberg relations for singular moduli

Yuqi Deng, Toshiki Matsusaka, Ken Ono

The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a na…

A discretization of the iterated integral expression of the multiple polylogarithm

Minoru Hirose, Toshiki Matsusaka, Shin-ichiro Seki

Recently, Maesaka, Watanabe, and the third author discovered a phenomenon where the iterated integral expressions of multiple zeta values become discretized. In this paper, we exte…

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

Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki +1

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…

math.NT2024★ 2 cited

On finite analogues of Euler's constant

Masanobu Kaneko, Toshiki Matsusaka, Shin-ichiro Seki

We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``…

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

Toshiki Matsusaka, Hideki Murahara, Tomokazu Onozuka

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,…

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

Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka

We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf-Witten invariant $Z_{k}$ for the ring of integers of the quadratic field $k = Q (p_{1} \dots p_{r})$ , where…

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

Rinnosuke Matsuhira, Toshiki Matsusaka, Koki Tsuchida

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. Althoug…