Zeta Function Research Articles

Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m2. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.Besides the Liouville action, at order m0, which only involves the background metric and the conformal factor σ, the various contributions to the effective gravitational action at higher orders in m2 can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m2, these contributions can be re-written as a term ∫ e2σσ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.

Quantum flux operators for Carrollian diffeomorphism in general dimensions

We construct Carrollian scalar field theories in general dimensions, mainly focusing on the boundaries of Minkowski and Rindler spacetime, whose quantum flux operators form a faithful representation of Carrollian diffeomorphism up to a central charge, respectively. At future/past null infinity, the fluxes are physically observable and encode rich information of the radiation. The central charge may be regularized to be finite by the spectral zeta function or heat kernel method on the unit sphere. For the theory at the Rindler horizon, the effective central charge is proportional to the area of the bifurcation surface after regularization. Moreover, the zero mode of supertranslation is identified as the modular Hamiltonian, linking Carrollian diffeomorphism to quantum information theory. Our results may hold for general null hypersurfaces and provide new insight in the study of the Carrollian field theory, asymptotic symmetry group and entanglement entropy.

Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincaré series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function.

A New Generalization of the Alternating Harmonic Series

Kilmer and Zheng (2021) recently introduced a generalized version of the alternating harmonic series. In this paper, we introduce a new generalization of the alternating harmonic series. A special case of our generalization converges to the Kilmer-Zheng series. Then we investigate several interesting and useful properties of this generalized, such as a summation formula related to the Hurwitz -Lerch Zeta function, a duplication formula, an integral representation, derivatives, and the recurrence relationship. Some important special cases of the main results are also discussed.

High-order SUSY-QM, the quantum XP model and zeroes of the Riemann Zeta function

Abstract Making use of the first- and second-order algorithms of supersymmetric quantum mechanics (SUSY-QM), we construct quantum mechanical Hamiltonians whose spectra are related to the zeroes of the Riemann Zeta function ζ(s). Inspired by the model of Das and Kalauni (DK) A Das and P Kalauni (2019 Physics Letters B 791, 265-269) which corresponds to this function in the strip 0 < Re [ s ] < 1 , and taking the factorization energy equal to zero, we use the wave function ∣x∣−S , S ∈ C , as a seed solution for our algorithms, obtaining XP-like operators. Thus, we construct SUSY-QM partner Hamiltonians whose zero energy mode locates exactly the nontrivial zeroes of ζ(s) along the critical line Re [ s ] = 1 / 2 in the complex plane. We further find that unlike the DK case, where the SUSY-QM partner potentials correspond to free particles, our partner potentials belong to the family of inverse squared distance potentials with complex couplings.

Zeta functions of p-adic analytic varieties

Zeta functions of p-adic analytic varieties

An improved explicit estimate for ζ(1/2 + it)

An explicit subconvexity bound for the Riemann zeta function ζ(s) on the critical line s=1/2+it is proved. This bound uses a new version of the Kusmin–Landau lemma and improves on current explicit bounds for |ζ(1/2+it)|.

Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet L-functions and those of spectral zeta functions or L-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values are bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.

L-functions of elliptic curves modulo integers

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see [9]). If E/K is an elliptic curve with nonconstant j-invariant defined over a function field K of characteristic p≥5, we know that its L-function L(T,E/K) is a polynomial in Z[T] (see [5, p. 11]). Inspired by Schoof, we study the reduction of L(T,E/K) modulo integers. We obtain three main results. Firstly, if E/K has non-trivial K-rational N-torsion for some positive integer N coprime with p, we extend a formula for L(T,E/K)modN due to Hall (see [4, p. 133, Theorem 4]) to all quadratic twists Ef/K with f∈K×∖K×2. Secondly, without any condition on the 2-torsion subgroup of E(K), we give a formula for the quotient modulo 2 of L-functions of any two quadratic twists of E/K. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and, under some assumption, find in most cases the exact analytic rank of these twists. We also illustrate that in favourable situations, our second main result allows one to compute much more efficiently L(T,Ef/K)mod2 than an algorithm of Baig and Hall (see [1]). Finally, we use our formulas to compute some degree 2 L-functions directly.

Zeta functions for tensor codes

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 [Formula: see text]-maximum rank distance codes and contains the [Formula: see text]-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.

Sums of infinite series involving the Riemann zeta function II

Sums of infinite series involving the Riemann zeta function II

On $\mu$-Zariski pairs of links

The notion of Zariski pairs for projective curves in $\mathbb{P}^{2}$ is known since the pioneer paper of Zariski. In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski) pair of curves $C = \{f(x,y,z) = 0\}$ and $C' = \{g(x, y, z) = 0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a Milnor number. We give new examples of weak Zariski pairs which have same $\mu^{*}$ sequences and same zeta functions but two functions belong to different connected components of $\mu$-constant strata (Theorem 14). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology, which implies the Jordan forms of their monodromies are different (Theorem 24). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that the hypersurface pair constructed from a Zariski pair of irreducible plane curves with simple singularities give a diffeomorphic links (Theorem 25).

Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models

AbstractWe consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the “zeta sector” of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e., K -theory of endomorphisms) of the “algebraic closure” of the absolute base \mathbb{S} . In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in one dimension and relate the Fourier transform (in one dimension) to its role over the algebraic closure of \mathbb{S} . We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

A note on odd zeta values over any number field and extended Eisenstein series

In this article, we study transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. Our transformation formula for the Dedekind zeta function at odd integers leads to a new number field extension of Eisenstein series, which satisfies the transformation z↦−1/z like an integral weight modular form over SL2(Z). The results provide a number of important applications, which are important in studying the behavior of odd zeta values as well as Lambert series in an arbitrary number field.

A unified approach to derive explicit solutions of generalized second-order linear recurrences and applications

We illustrate the versatility of our previous work on closed-form solutions of second-order linear recurrences by specializing the underlying counting sets to prove new identities. Extensions allowing the sequence values to be functions of a variable t as well as the integer variable n are mentioned and applied to derive closed-form solutions, both classical and new. Connections with continued fractions are established and used to settle several conjectures from the Ramanujan Machine. Finally, alternative verifications of well-known continued fractions such as those of zeta functions and continued fractions in Ramanujan’s second notebook are presented.

Casimir free energy for massive scalars: A comparative study of various approaches

We compute the Casimir thermodynamic quantities for a massive real scalar field between two parallel plates with the Dirichlet boundary conditions, using three different general approaches and present explicit solutions for each. The Casimir thermodynamic quantities include the Casimir Helmholtz free energy, pressure, energy, and entropy. The three general approaches that we use are based on the fundamental definition of Casimir thermodynamic quantities, the analytic continuation method, and the zero temperature subtraction method. Within the analytic continuation approach, we use two distinct methods which are based on the utilization of the zeta function and the Schlömilch summation formula. We include the renormalized versions of the latter two approaches as well, whereas the first approach does not require one. Within each general approach, we obtain the same results in a few different ways to ascertain the selected cancellations of infinities have been done correctly. We show that, as expected, the results based on the zeta function and the Schlömilch summation formula are equivalent. We then do a comparative study of the three different general approaches and their results and show that they are in principle not equivalent to each other, and they yield equivalent results only in the massless case. In particular, we show that the Casimir energy calculated only by the first approach has all three properties of going to zero as the temperature, mass of the field or the distance between the plates increases. Moreover, we show that in this approach the Casimir entropy reaches a positive constant in the high temperature limit, which can explain the linear term in the Casimir free energy. We use the results obtained by the fundamental approach as our reference for the correct results.

On the properties of invariant functions

If f(x,y) is a real function satisfying y>0 and ∑r=0n−1f(x+ry,ny)=f(x,y) for n=1,2,3,…, we say that f(x,y) is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz zeta function are related to invariant functions. In this paper we systematically investigate the properties of invariant functions.

Global series for height 1 multiple zeta functions

Global series for height 1 multiple zeta functions

On cohomology of generalized Suzuki curves and exponential sums

We study the étale cohomology of generalized Suzuki curves as representations of their automorphism groups. We determine exact values of certain exponential sums related to the curves. As a result, we calculate numbers of rational points and the zeta functions of the curves. As an application, we find several maximal curves.