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Toshiki Matsusaka

Publications and source records attributed to Toshiki Matsusaka.

At least 19 recordsLinked to original sources
arXiv2026

Distribution of Kaneko's val function

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 hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoglu to construct a bounded continuous observable.

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arXiv2026

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

This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal{A}$. In the ring $\mathcal{A}$, one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results of Anzawa-Funakura and Luca-Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as $\log_{\mathcal{A}}(2)$ under the ABC conjecture.

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arXiv2026

Poly-Bernoulli numbers from shifted log-sine integrals

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this function essentially by design, it is natural to ask whether they arise as special values of more intrinsic zeta-type objects. In this article, we show that a shifted log-sine integral provides such an example. Its analytically continued values at negative integers are given by anti-diagonal sums of poly-Bernoulli numbers with negative index.

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arXiv2025

On the Alexander polynomials of modular knots

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to Lorenz knots and following the notion of modular knots introduced by Ghys, this article investigates the topological $\mathrm{SL}_2(\mathbb{Z})$-invariants arising from modular knots. Our main focus is the Alexander polynomial of modular knots. Using the Burau representation, we highlight two contrasting features of this family. On the one hand, for each fixed degree, only finitely many Alexander polynomials of modular knots occur. On the other hand, any integer appears as a coefficient of the Alexander polynomial of some modular knot, and coefficients of the same sign can occur in runs of arbitrarily long length.

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arXiv2025

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

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 presented as a consequence of Zagier's work inspired by Borcherds products. Since then, the formula has developed into a broader framework that links the Fourier coefficients of modular forms to the special values of modular functions, extending in various directions. This article surveys these subsequent developments.

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arXiv2025

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

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 series identities for some powers of $\triangle(q)$, where $\triangle(q)$ is the generating function of triangular numbers. In this article, we give a different proof of one of their identities. The main step is to prove that a certain indefinite theta function involving spherical polynomials is a modular form. We use the technique recently developed by Roehrig and Zwegers.

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arXiv2025

Applications of Faà di Bruno's formula to partition traces

We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Faà di Bruno formula. This approach offers a unified and natural reinterpretation of known results and provides a systematic framework for deriving new identities of a similar type.

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arXiv2025

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

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 power series identities for powers of the generating function of triangular numbers. We also show that these identities arise as specializations of denominator identities of affine Lie superalgebras.

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arXiv2025

Hecke equivariance of the divisor map

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 applications, we investigate the divisor sum formula of Bruinier-Kohnen-Ono and more general Rohrlich-type divisor sums for polyharmonic Maass forms, discussing several implications for the Hecke action and its relation to the self-adjointness of the Hecke operators.

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arXiv2025

On finite analogues of Euler's constant

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 ``regularized value of $ζ(1)$'' and of Mascheroni's and Kluyver's series expressions of Euler's constant using Gregory coefficients. Moreover, we reveal that the differences between them always lie in the $\mathbb{Q}$-vector space spanned by 1 and values of a finite analogue of logarithm at positive integers.

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arXiv2024

A unified approach to Rohrlich-type divisor sums

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, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich's formula to level $N$, (3) an explicit version of the theorem by Bringmann-Kane-Löbrich-Ono-Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann-Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.

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arXiv2024

Hauptmoduln and even-order mock theta functions modulo 2

The Fourier coefficients $c_1(n)$ of the elliptic modular $j$-function are always even for $n \not\equiv 7 \pmod{8}$. In contrast, for $n \equiv 7 \pmod{8}$, it is conjectured that ``half" of the coefficients take odd values. In this article, we first observe in detail when $c_1(8n-1)$ is odd and show that the coefficients share the same parity as the coefficients $c_{μ_2}(n)$ of the 2nd order mock theta function $μ_2(q)$. Furthermore, we prove that this phenomenon also holds among several hauptmoduln and between hauptmoduln and even-order mock theta functions.

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arXiv2024

Eichler-Selberg relations for singular moduli

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 natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(τ)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m \left(j(τ)-744\right). \] For each $ν\geq 0$ and $m\geq 1$, we obtain an Eichler-Selberg relation. For $ν=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(τ).$ For each $ν\geq 1$ and $m\geq 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2ν+2$ cusp forms, where the traces of $j_m(τ)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions.

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arXiv2024

A discretization of the iterated integral expression of the multiple polylogarithm

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 extend their result to the case of multiple polylogarithms and provide two proofs. The first proof uses the method of connected sums, while the second employs induction based on the difference equations that discrete multiple polylogarithms satisfy. We also investigate several applications of our main result.

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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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