Spectral zeta function on discrete tori and Epstein-Riemann conjecture

Discrete sums of classical symbols on Z^d and zeta functions associated with Laplacians on tori

We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points of infinite unions of nested convex polytopes in $\R^d$. This canonical regularised sum is the building block to construct meromorphic extensions of the ordinary sum on holomorphic symbols. Explicit formulae for the complex residues at their poles are given in terms of noncommutative residues of classical symbols, thus extending results of Guillemin, Sternber…

Determinant of Laplacian on tori of constant positive curvature with one conical point

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.

正交平坦环面上Zeta正则化行列式的全局极值

Global Extrema of the Zeta Regularized Determinant on Orthogonal Flat Tori

研究单位体积正交平坦环面中最大化Laplacian Zeta正则化行列式的问题，证明等边环面是唯一极大值点，验证了Sarnak猜想，并探讨了Neumann和Dirichlet边界条件下的类似问题。

Length orthospectrum and the correlation function on flat tori

This note presents some of the results obtained in arXiv:2207.05410 and it has beenthe object of a talk of the second author during the Journées "Équations auxDérivées Partielles" (Obernai, june 2022). We study properties of geodesics that are orthogonal to two convex subsets of the flat torus. We discuss meromorphic properties of a geometric Epstein zeta function associated to the set of lengths of such orthogeodesics. We also define the associated length distribution and discuss singularities of its Fourier transform. Our analysis relies on a fine study of the dynamical correlation function …

Regularized limit of determinants for discrete tori

We consider a combinatorial Laplace operator on a sequence of discrete graphs which approximates the m-dimensional torus when the discretization parameter tends to infinity. We establish a polyhomogeneous expansion of the resolvent trace for the family of discrete graphs, jointly in the resolvent and the discretization parameter. Based on a result about interchanging regularized limits and regularized integrals, we compare the regularized limit of the log-determinants of the combinatorial Laplacian on the sequence of discrete graphs with the logarithm of the zeta determinant for the Laplace Be…

具有收敛性质的离散环面上的联络拉普拉斯算子

Connection Laplacian on discrete tori with converging property

该研究分析实环面与离散环面上联络拉普拉斯算子的谱性质，提出基于平行正交基的研究方法，证明离散环面经尺度变换的特征值收敛于实环面特征值，并推导了正则化对数行列式的相关公式。