Related papers: Zeta function factorisation, Dwork hypersurfaces, …

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard--Fuchs differential equations; we count points using Gauss sums and…

Standard zeta function regularisation enforces a scale-independent prescription for spectral aggregation, effectively fixing the relative weight of spectral contributions. We relax this constraint by replacing the derivative at $s=0$ with a…

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

Let $\Theta_{3} (z):= \sum_{n\in\mathbb{Z}} \exp (i \pi n^2 z)$ be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane $\mathbb{H}$, and takes positive values along the positive imaginary axis. We…

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the…

We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using $p$-adic cohomology. This includes new bounds for the $p$-adic and $t$-adic precisions…

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We…

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and…

In this paper, we study two compactifications of general hypersurfaces defined by the vanishing of linear combinations of $d+2$ monomials in $d$-dimensional algebraic tori. We prove that the number of their $\mathbb{F}_q$-points is given by…

We define supersymmetric zeta functions and supersymmetric determinants, which can reveal spectral properties complementary to those captured by the supersymmetric indices. They play a crucial role in analyzing the Cardy-like behaviors of…

We study the special value conjecture for the Zeta function of a proper regular arithmetic scheme X introduced by Flach and Morin in the case n=1. We compute the correction factor C(X,1) left unspecified in the original statement of the…

We revisit the factorisation of supersymmetric partition functions of 3d $\mathcal{N}=4$ gauge theories. The building blocks are hemisphere partition functions of a class of UV $\mathcal{N}=(2,2)$ boundary conditions that mimic the presence…

The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic…

We explicitly give factorization formulas for higher depth determinants, which are defined via derivatives of the spectral zeta function at non-positive integer points, of Laplacians on the n-sphere in terms of the multiple gamma functions.

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of…

For the Tits building B(G) of a finite group of Lie type G(Fq), we study the edge zeta function, which enumerates edge-geodesic cycles in the 1-skeleton. We show that every nonzero edge eigenvalue becomes a power of q after raising to a…

This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.

We present a new expansion of the zeta-function of Riemann. The current formalism -- which combines both the idea of interpolation with constraints and the concept of hypergeometric functions -- can, in a natural way, be generalised within…