Narrow Results

We give two applications of an explicit formula for global zeta functions of prehomogeneous vector spaces in Math. Ann.315 (1999), 587–615. One is concerned with an explicit form of global zeta functions associated with Freudenthal quartics, and the other the comparison of the zeta function of a unsaturated prehomogeneous vector space with that of the saturated one obtained from it.

We introduce a generalized Mahler measure. It has relations to multiple sine functions and Dirichlet L-functions. In particular, we are able to express special values of Dirichlet L-functions by sum of logarithmic generalized Mahler measures.

We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarized abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in P⁴ which is the complete intersection of two S₅-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.

We use the generalized Chowla–Selberg formula to consider the Casimir effect of a scalar field with a helix torus boundary condition in the flat (D + 1)-dimensional spacetime. We obtain the exact results of the Casimir energy density and pressure for any D for both massless and massive scalar fields. The numerical calculation indicates that once the topology of spacetime is fixed, the ratio of the sizes of the helix will be a decisive factor. There is a critical value r_c of the ratio r of the lengths at which the pressure vanishes. The pressure changes from negative to positive as the ratio r passes through r_c increasingly. In the massive case, we find the pressure tends to the result of massless field when the mass approaches zero. Furthermore, there is another critical ratio of the lengths and the pressure is independent of the mass at in the D = 3 case.

We compute the Casimir energy which arises in a bi-dimensional surface due to the quantum fluctuations of a scalar field. We assume that the boundaries are non-flat and the field obeys Dirichlet condition. We re-parametrize the problem to one which has flat boundary conditions and the irregularity is treated as a perturbation in the Laplace–Beltrami operator associated to the coordinate transformation. Later, to compute the Casimir energy, we use zeta function regularization. It is compared with the results coming from perturbation theory with the one from Wentzel–Kramers–Brillouin (WKB) method.

For the three-dimensional BTZ black hole we consider a Selberg type zeta function. We indicate how special values of its logarithm correspond to certain thermodynamic quantities associated with the black hole.

This work investigates zeta functions for d-dimensional shifts of finite type, d ≥ 3. First, the three-dimensional case is studied. The trace operator T_a1,a2;b12 and rotational matrices R_x;a1,a2;b12 and R_y;a1,a2;b12 are introduced to study -periodic patterns. The rotational symmetry of T_a1,a2;b12 induces the reduced trace operator τ_a1,a2;b12 and then the associated zeta function ζ_a1,a2;b12 = (det(I-s^a₁a₂τ_a1,a2;b12))^-1. The zeta function ζ is then expressed as , a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL₃(ℤ). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d ≥ 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)={\sum }_{n\in \mathbb{N}}{a}_n(G)/n^s$ is associated, where $a_n(G)={\sum }_{|G:H|=n}\mu (H,G)$ and $\mu (H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse ${(P_G(s))}^{-1}$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of ${(P_G(s))}^{-1}$ is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

This paper considers non-commutative curves, introduces a divisor class group and a degree map, proves a Riemann-Roch theorem, and solves the Riemann-Roch problem. These results are then used to prove the zeta function of a non-commutative curve over a finite field satisfies the two first Weil conjectures.

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of $j$-maximum rank distance codes and contains the $j$-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s_n = 1/2 + iλ_n. By constructing a continuous family of scaling-like operators involving the Gauss–Jacobi theta series and by invoking a novel -invariant Quantum Mechanics, involving a judicious charge conjugation and time reversal operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the "fibration condition" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.

A vanishing theorem is proved for ℓ-adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibility of the poles or equivalently an improvement of the polar part of the Ax-Katz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums.

It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζ_k(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

Let ${𝔽}_q$ be a finite field, $F/{𝔽}_q$ be a function field of genus $g$ having full constant field ${𝔽}_q$, ${𝒮}$ a set of places of $F$ and $H$ the holomorphy ring of ${𝒮}$. In this paper, we compute the density of coprime $m$-tuples of elements of $H$. As a side result, we obtain that whenever the complement of ${𝒮}$ is finite, the computation of the density can be reduced to the computation of the $L$-polynomial of the function field. In the genus zero case, classical results for the density of coprime $m$-tuples of polynomials are obtained as corollaries.

Let $m$ be a positive integer, and define

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.