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In this paper, we study the Laurent coefficients of meromorphic modular forms at CM points by giving two approaches of computing them. The first is a generalization of the method of Rodriguez-Villegas and Zagier, which expresses the Laurent…
A formal definition of the graded algebra $\mathcal{R}$ of modular linear differential operators is given and its properties are studied. An algebraic structure of the solutions to modular linear differential equations (MLDEs) is shown. It…
In this note we report on the conjectural identification of paramodular forms from Calabi-Yau motives of Hodge type (1, 1, 1, 1) of moderately low conductor. We calculate Euler factors from Calabi-Yau operators from the AESZ database by the…
In this paper, we use the Rogers-Ramanujan type $q$-exponential operator $\mathcal{R}(qD_{q})$ to derive generating functions, and Mehler and Rogers formulas, for the non-normalized homogeneous Stieljes-Wigert polynomials…
In this paper, we show some expressions of certain $q$-multiple zeta-star values at roots of unity. These explicit formulas are expressed by using the determinants or Bell polynomials. Explicit formulas for other types of values can be…
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…
We compute four-denominator angular phase-space integrals using the Mellin--Barnes (MB) technique in dimensional regularisation. Independent of the scattering process, an angular integral can be categorised based on the nature of the…
We study the robust output regulation of linear boundary control systems by constructing extended systems. The extended systems are established based on solving static differential equations under two new conditions. We first consider the…
We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+\epsilon})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the…
We present a formula for the regulator of two arbitrary Siegel units in terms of L-values of pairwise products of Eisenstein series of weight one. We give applications to Boyd's conjectures and Zagier's conjectures for elliptic curves of…
We prove a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is…
We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the Mahler measures and using…
We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's…
Boolean calculus has been studied extensively in the past in the context of switching circuits, error-correcting codes etc. This work generalizes several approaches to defining a differential calculus for Boolean functions. A unified theory…
In this paper we introduce a class of determinants "of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms.
The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in…
Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let ${\rm det}\,\Delta$ be the $\zeta$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for ${\rm…
We calculate a Zamolodchikovs' triple integral by the Bernstein-Reznikov method.
Analytic continuation and functional equation of a Dirichlet series constructed from two (not necessarily cuspidal) holomorphic modular forms is discussed, where either weights of the modular forms or characters are not necessarily equal to…
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated…