The periodic zeta-function $\zeta(s; a)$, $s = \sigma + it$, $a = \{a_m \in \mathbb{C} : m \in \mathbb{N}\}$, in the half-plane $\sigma > 1$ is defined by Dirichlet series with periodic coefficients $a_m$, and has the meromorphic continuation to the whole complex plane. The function $\zeta(s; a)$ is a generalization of the Riemann zeta-function and Dirichlet $L$-functions. In the paper, using only the periodicity of the sequence $a$, we obtain that the shifts $\zeta(s + i\tau; a)$, $\tau \in \mathbb{R}$, approximate a certain class of analytic functions, defined in the strip $\{s \in \mathbb{C} : 1/2 < \sigma < 1\}$. For $T^{23/70} \leqslant H \leqslant T^{1/2}$, the set of such shifts has a positive lower density in the interval $[T, T + H]$, $T \to \infty$. The case of positive density is also discussed. For the proof, the mean square estimate in short intervals for the Hurwitz zeta-function, and probabilistic limit theorems are applied.

This paper explains why the critical line sits at the real part equal to one-half by treating it as an intrinsic boundary of a reparametrized complex plane (“z-space”), not a mere artifact of functional symmetry. In z-space the real part is defined by a geometric-series map that gives rise to a rulebook for admissible analytic operations. Within this setting we rederive the classical toolkit—the eta–zeta relation, Gamma reflection and duplication, theta–Mellin identity, functional equation, and the completed zeta—without importing analytic continuation from the usual s-variable. We show that access to the left half-plane occurs entirely through formulas written on the right, with boundary matching only along the line with the real part equal to one-half. A global Hadamard product confirms the consistency and fixed location of this boundary, and a holomorphic change of variables transports these conclusions into the classical setting.

We study the global in time existence of small solutions to the Cauchy problem for the subcritical modified Korteweg-de Vries (KdV) equation [Formula: see text] We suppose that [Formula: see text] . We remark that [Formula: see text] means that equation is subcritical in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small. Then we find the large time asymptotics of the solutions.

In this paper, we first study analytic properties of Koecher–Maass series associated to Hermitian modular forms. Next, we define certain kernel function associated to Hermitian modular forms and modular forms on the half-space of quaternions and study their analytic properties. As an application, we derive another proof of analytic continuation and functional equation of the Koecher–Maass series associated to these modular forms.

Analytic extensions of 𝐴_{∞}-weights on Lipschitz curves and their use in weighted Hardy spaces

A steady-state model for tilting pad journal bearings incorporating an analytical extension of the Sommerfeld solution

This study develops a rigorous analytic framework for solving the Cauchy problem of polyharmonic equations in , highlighting the crucial role of symmetry in the structure, stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions of Laplace and biharmonic equations, frequently arise in elasticity, potential theory, and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality reflects the underlying rotational and reflectional symmetries, the study constructs explicit, uniformly convergent series solutions. Through analytic continuation of integral representations, necessary and sufficient solvability criteria are established, ensuring convergence of all derivatives on compact domains. Furthermore, newly derived Green-type identities provide a systematic method to reconstruct boundary information and enforce stability constraints. This approach not only generalizes classical Laplace and biharmonic results to higher-order polyharmonic equations but also demonstrates how symmetry governs boundary data admissibility, convergence, and analytic structure, offering both theoretical insights and practical tools for elasticity, inverse problems, and mathematical physics.

Phenomenological relativistic models of strongly interacting systems have proved to be useful for modeling few-nucleon and few-quark systems at the few-GeV scale. They have the advantage that relativistic invariance is exact; however, satisfying cluster properties requires a complicated construction which has never been used in applications. In addition, there is no straightforward connection to local field theories. To explore an alternative approach for constructing relativistic quantum mechanical models that addresses these two deficiencies. This work examines the general structure of the dynamical input to these models. Relativistic models that address these two deficiencies can be formulated using reflection positive Euclidean kernels. Relaxing the locality requirement of the axioms of Euclidean field theories leads to a weaker form of reflection positivity. Constraints imposed by the spectral condition and cluster properties are used to determine the structure of a class of kernels satisfying this weaker form of reflection positivity. The resulting kernels lead to a representation of the quantum mechanical Hilbert space in terms of Euclidean variables that has a positive norm, a set of self-adjoint Poincaré generators with a Hamiltonian satisfying cluster properties and a spectral condition. Relativistic dynamical calculations can be performed directly in the Euclidean representation without analytic continuation. Schwinger functions of a local field theory also satisfy the weakened form of reflection positivity, which provides the desired connection with local field theories. This work provides the structure of a large class of reflection positive kernels that can be used to test this Euclidean formulation of relativistic quantum mechanics. What is still missing is a dynamical principle that generates model kernels with the desired properties.

The Riemann zeta function (s) occupies a fundamental role in analytic number theory. This review covers its primary attributes, initiating with the analytic extension past the convergence area (s) > 1. The functional equation receives special focus, illustrating the functions symmetry and its evaluations at distinct locations, particularly negative integers. The study delves into the association of (s) with the theta function through the Mellin transform, revealing connections to modular forms. Representation theory from the Heisenberg group further elucidates the zeta functions spectral and harmonic elements in automorphic forms. The examination concludes with uses in prime number theory, demonstrating (s)s contribution to detailed prime distribution models. These aspects together highlight the zeta functions extensive relevance in number theory and associated areas

In the inverse Stieltjes moment problem, one seeks to reconstruct a non-negative distribution from its spectral moments defined on an unbounded interval 〈0,∞. In chemical physics, this problem arises when computing continuous quantities such as photoionization cross sections or electronic decay widths using discretized approximations to the electronic continuum. While Stieltjes imaging (SI) is the established method in this context, it provides only sparse, discrete sampling of the distribution. Here, we develop a maximum entropy (ME) approach to the solution of the inverse Stieltjes moment problem in the context of Fano theory of resonances, where the sought-after quantity is the decay width function. We implement two ME variants-with polynomial and exponential asymptotic damping-and introduce an averaging procedure over spectral moment orders that addresses convergence issues and provides reliable error estimates. Benchmarking against abinitio Fano-ADC data for molecular Auger decay and interatomic Coulombic decay, we show that ME achieves comparable accuracy to SI while providing a continuous representation of the function. Our results establish ME as a valuable alternative to SI, particularly when analytical continuation or additional verification is required.

A bstract The time-ordered multilayer integrals have long been cited as major challenges in the analytical study of cosmological correlators and wavefunction coefficients. The recently proposed family tree decomposition technique solved these time integrals in terms of canonical objects called family trees, which are multivariate hypergeometric functions with energies as variables and twists as parameters. In this work, we provide a systematic study of the analytical properties of family trees. By exploiting the great flexibility of Mellin representations of family trees, we identify and characterize all their singularities in both variables and parameters and find their exact series representations around all singularities with finite convergent domains. These series automatically generate analytical continuation of arbitrary family trees over many distinct regions in the energy space. As a corollary, we show the factorization of family trees at zero partial-energy singularities to all orders. Our findings offer essential analytical data for further understanding and computing cosmological correlators.

A bstract We investigate conformal field theories with gauge group U( N ) at arbitrary rank N , focusing on the role of trace relations in determining the structure of the Hilbert space. Working in the free trace algebra without imposing relations, we identify a class of evanescent states that vanish at finite N . Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge Q b . This framework allows us to define and compute transition amplitudes between evanescent and physical states, which we show correspond precisely to ordinary CFT amplitudes analytically continued in N . Our results provide a direct algebraic realization of the proposals which realize trace relations in the bulk as over-maximal giant gravitons [1–3] and establish analytic continuation in N as a powerful tool for understanding finite- N effects.

A bstract This work investigates holographic timelike entanglement entropy in higher curvature gravity, with a particular focus on Lovelock theories and on the role of excited states. For strip subsystems, higher-curvature terms are found to affect the imaginary part of the entropy in a dimension-dependent manner, while excited states contribute solely to the real part. For the cases analyzed, spacelike and timelike entanglement entropies exhibit proportional relations: vacuum contributions differ by universal phase factors, while excitation contributions are linked by dimension-dependent rational coefficients. For hyperbolic subsystems, the timelike entanglement entropy computed via complex extremal surfaces is shown to agree with results obtained through analytic continuation, with imaginary contributions appearing in all dimensions. Higher-curvature corrections are explicitly calculated in five- and ( d + 1)-dimensional Gauss-Bonnet gravity, illustrating the applicability of the complex surface prescription to general Lovelock corrections. These results provide a controlled setting to examine the influence of higher-curvature interactions on holographic timelike entanglement entropy, and clarify its relation to vacuum and excited-state contributions.

The complex analytic continuation can be developed to enable the Gauss-Krüger projection to be non-zonal and non-singular in polar regions. The series expansion of the traditional Gauss-Krüger projection in terms of third flattening has been derived by using a computer algebra system, leading to a substantial simplification of the final formulas without compromising accuracy compared with the series expansion in terms of eccentricity. Therefore, the non-zonal formulas of the Gauss-Krüger projection in term of third flattening have been expressed, and the non-zonal and the non-singular formulas of the Gauss-Krüger projection has been derived by the conformal colatitude. With respect to the mapping of 4 high-latitude regions (Finland, Sweden, Norway and Alaska) and its isopleth map, it was verified that the non-zonal and the non-singular algorithm of the Gauss-Krüger projection had high precision and minimal distortion in polar regions. The method presented a meaningful supplement to the existing Gauss–Krüger projection family.

The paper considers the problem of analytical continuation of special functions by branched continued fractions. These representations play an important role in approximating of special functions that arise in various applied problems. By improving the methods of studying the convergence of branched continued fractions, several domains of analytical continuation of the special function $H_4(\alpha,\delta+1;\gamma,\delta;-\mathbf{z})/H_4(\alpha,\delta+2;\gamma,\delta+1;-\mathbf{z})$ in the case of real and complex parameters are established. To prove the analytical continuation, the so-called PC method is used, which is based on the principle of correspondence between a formal double power series and a branched continued fraction. An example is provided at the end.

Abstract This paper addresses the problem of recovering the spatial profile of the source in the complex Ginzburg–Landau equation (CGLE) from regional observation data at fixed times. We establish two types of sufficient measurements for the unique solvability of the inverse problem. The first is to determine the source term by using whole data at one fixed instant. Conditional stability is established by using the eigenfunction expansion argument. Next, using the analytic continuation method, both uniqueness and a stability estimate for recovering the unknown source can be established from local data at two instants. Finally, to effectively handle the complex-valued solutions of the CGLE, we propose a novel complex physics-informed neural networks (C-PINNs) framework. This approach designs complex-valued layers that inherently respect the complex structure of the governing equation, overcoming limitations of standard real-valued PINNs for such dissipative systems. Numerical experiments demonstrate the accuracy and efficiency of our C-PINNs algorithm in recovering the source term.

On the Domain of Analytic Extension of Lauricella–Saran’s Hypergeometric Functions FM and their Ratios

Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism - separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians - thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron-phonon models. 
The key conceptual advance is the nature index η, which complements the conventional statistical sign ζ. While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles (η = 0) from second quantization of systems with massive particles (η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. 
From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results - such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron-phonon systems - are either absent from, or only sparsely
mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. 
Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.

Let G G be a finite abelian p p -group. We count étale G G -extensions of global rational function fields F q ( T ) \mathbb F_q(T) of characteristic p p by the degree of what we call their Artin–Schreier conductor. The corresponding (ordinary) generating function turns out to be rational. This gives an exact answer to the counting problem, and seems to beg for a geometric interpretation. This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane.

The paper considers the problem of the analytical extension of the ratios of generalized hypergeometric functions 3F2. A new domain of analytic continuation for these ratios under certain conditions to parameters is established. In this case, the domain of analytic extension of the special function is the domain of convergence of its branched continued fraction expansion. This paper also provides an example of applying the obtained results to dilogarithm function.