Related papers: The sigma function for trigonal cyclic curves

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel-Jacobi map, which allows us to identify the…

In this paper we study the Hilbert function of $\gr_{\mathfrak{m}}(R)$, when $R$ is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical…

We present a compact formula for the supersymmetric partition function of 2d N=(2,2), 3d N=2 and 4d N=1 gauge theories on $\Sigma_g \times T^n$ with partial topological twist on $\Sigma_g$, where $\Sigma_g$ is a Riemann surface of arbitrary…

We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably…

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current…

We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.

Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus $1$ sigma-function and elementary functions as solutions of a system of linear PDEs satisfied by the sigma-function. By way of application we derive a…

This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an…

The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…

We refine a previous work of K. Matsumoto and H. Ishikawa, obtaining an asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip (1/4<$\sigma$<1/2), by obtaining an…

Let $\Sigma_g$ be a closed Riemann surface of genus $g$. Let $G$ be a finite subgroup of the automorphism group of $\Sigma_g$. It is well known that there exists a smooth $G$-equivariant embedding from $\Sigma_g$ to some Euclidean space…

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not…

A cyclic trigonal curve of genus three is a $\mathbb{Z}_3$ Galois cover of $\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass for the…

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

The Riemann $\Xi(z)$ function (even in $z$) admits a Fourier transform of an even kernel $\Phi(t)=4e^{9t/2}\theta''(e^{2t})+6e^{5t/2}\theta'(e^{2t})$. Here $\theta(x):=\theta_3(0,ix)$ and $\theta_3(0,z)$ is a Jacobi theta function, a…