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We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in P 4 . Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H 3 (W , Z) . It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ 0 (25) , whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M /ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel's original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that "half" of Siegel's zeta functions correspond to holomorphic modular forms. [ABSTRACT FROM AUTHOR]

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as q → 1 , we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be 'lifted' to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula for the sum where the run over the imaginary parts of nontrivial zeros of the Riemann zeta function with multiplicities taken into account and the function belongs to some special class of functions in . [ABSTRACT FROM AUTHOR]

We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. [ABSTRACT FROM AUTHOR]

We show that the Generalized Riemann Hypothesis for Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of ζ (s) satisfy those properties under GRH. [ABSTRACT FROM AUTHOR]

We prove that the Riemann zeta-function ζ (σ + i t) has no zeros in the region σ ≥ 1 - 1 / (55.241 (log | t |) 2 / 3 (log log | t |) 1 / 3 ) for | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region σ ≥ 1 - 1 / (5.558691 log | t |) for | t | ≥ 2 . We also provide new bounds that are useful for intermediate values of | t | . Combined, our results improve the largest known zero-free region within the critical strip for 3 · 10 12 ≤ | t | ≤ exp (64.1) and | t | ≥ exp (1000) . [ABSTRACT FROM AUTHOR]

We construct some class of selfadjoint operators in the Krein spaces consisting of functions on the straight line . Each of these operators is a rank-one perturbation of a selfadjoint operator in the corresponding Hilbert space and has eigenvalues complex numbers of the form , where ranges over the set of nontrivial zeros of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

We derive the exact form of the spectral interaction of two strings mediated by a constant scalar field using methods derived from noncommutative geometry. This is achieved by considering a non-product modification of the Connes–Lott model with two-dimensional manifolds. The analogy with the latter construction justifies the interpretation of the scalar field as being of Higgs type. Working in dimension two requires the use of the spectral zeta function instead of the Wodzicki residue techniques applicable to four-dimensional models. In the latter case, an analogous non-product geometry construction leads, for specific choices of metrics, to the so-called "doubled geometry models", which can be thought of as a spectral modification of the Hassan–Rosen bimetric theory. We find that in dimension two, the interaction term depends explicitly on zweibeins defining the Dirac operators and only in some special cases can they be expressed solely using the metrics. The computations can be performed analytically for an arbitrary choice of zweibeins defining geometry on the two strings. [ABSTRACT FROM AUTHOR]

By applying the "coefficient extraction method" to hypergeometric series, we establish several remarkable infinite series identities about harmonic numbers and five binomial coefficients, including three conjectured by Z.-W. Sun. [ABSTRACT FROM AUTHOR]

This article, titled "Invariant for Parameter-Dependent Boundary Value Problems," discusses the concept of -invariants in mathematics. The -invariant is a spectral invariant that represents the contribution of certain points to the index of elliptic operators. The article focuses on the construction of the -invariant for parameter-dependent boundary value problems and presents the main results and theorems related to this topic. It also provides an example to illustrate the calculations involved in determining the -invariant. The work was supported by the Young Russian Mathematics award and the Russian Foundation for Basic Research. [Extracted from the article]

This note gives some applications of the zeta distribution to number theory. First, a number of probabilistic proofs of the equality between the Riemann zeta functions and some number-theoretic functions are provided. Second, some problems of probability with respect to positive integers considered in the literature are discussed in the zeta distribution case. In contrast to usual heuristic arguments of probabilistic approach in number theory where the uniform distribution is assumed on positive integers, the proofs here are rigorous by utilizing the zeta distributions on positive integers. [ABSTRACT FROM AUTHOR]

The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator originally introduced as a generalization of the quantum harmonic oscillator having a weaker sl 2 (R) -symmetry. The spectrum of the NCHO has remarkable properties, including the presence of number theoretical structures such as modular forms, elliptic curves and Eichler cohomology observed in the special values of the associated spectral zeta function. In addition, the Heun ODE picture of the eigenvalue problem of the NCHO reveals a connection with the quantum Rabi model (QRM), a fundamental interaction model from quantum optics. In this paper we introduce and describe the basic properties of an η -shifted NCHO (η -NCHO) that has an analogous relation through the respective Heun ODE pictures with the asymmetric quantum Rabi model (AQRM) which breaks an apparent symmetry of the QRM. Even though the shift factor does not break the parity symmetry of the NCHO, a certain type of degeneracies appears for η ∈ 1 2 Z , as if mirroring the situation of the AQRM. We give furthermore a detailed description of the confluence process, that we call iso-parallel confluence process due to the fact that it requires a parallel transformation of two parameters describing the spectrum of η -NCHO and representations of sl 2 (R) . We relate the eigenvalues of the two models under the iso-parallel confluence process, including how the quasi-exact eigenfunctions of the η -NCHO correspond to Juddian solutions of the AQRM. From the point of view of this confluence process, a family of η -NCHO corresponds to a single AQRM, thus we may regard the η -NCHO as a covering of the AQRM. We expect the study of the η -NCHO and the AQRM from this point of view to be helpful for the clarification of several questions on the AQRM, including the hidden symmetry and the number of Juddian solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 = 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. [ABSTRACT FROM AUTHOR]

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest. [ABSTRACT FROM AUTHOR]

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p -adic (local and global) Langlands correspondence for GL 2 / Q . As an application of our zeta morphisms and the recent article (Kim et al. in On the Iwasawa invariants of Kato's zeta elements for modular forms, 2019, arXiv:1909.01764v2), we prove a theorem which roughly states that, under some μ = 0 assumption, Iwasawa main conjecture without p -adic L -function for f holds if this conjecture holds for one g which is congruent to f . [ABSTRACT FROM AUTHOR]

We study the spin- relativistic fermions influenced by the Dirac oscillator in Rindler's space–time. The energy eigenvalues of this oscillator enable us to calculate the thermodynamic properties of this oscillator by using the Hurwitz zeta function via the Mellin transformation. The effect of the geometry of space–time on these properties is studied. [ABSTRACT FROM AUTHOR]

For a number field K , the Euler–Kronecker constant γ K associated to K is an arithmetic invariant the size and nature of which is linked to some of the deepest questions in number theory. This theme was given impetus by Ihara who obtained bounds, both unconditional as well as under GRH for Dedekind zeta functions. In this note, we study the analogous constants associated to the narrow ray class fields of an imaginary quadratic field. Our goal for studying such families is twofold. First to show that for such families, the conditional bounds obtained by Ihara can be improved on the average, again under GRH for Dedekind zeta functions. Further, our family of number fields are non-abelian while such average bounds have earlier been studied for cyclotomic fields. The technical part of our work is to make the dependence of these upper bounds on the ambient number field explicit. Such explicit dependence is essential to further our objective. [ABSTRACT FROM AUTHOR]

We consider the following two problems. We are given the values of several initial derivatives of the Riemann zeta function calculated at some (unknown to us) point a. • How could we calculate an approximate value of the function itself at the same point a without prior finding this number? • How could we find an approximate value of a itself? We suggest several algorithms for answering these questions and demonstrate their accuracy on a few numerical examples. The algorithms reveal some new properties of the zeta function. [ABSTRACT FROM AUTHOR]

Gijswijt's Sequence (A090822) For this sequence, it will be helpful to remember that chemists do not write HT ht for water; they write HT ht . So the sequence with a single term 99, say, can be written as HT ht , where I Y i is the number 99, and it has curling number 1. If you think of an animal with its head looking to the left, with a very curly tail, then I X i represents the head and body of the animal, and HT ht represents the curls in its tail. The case of an HT ht grid seems even harder. [Extracted from the article]

We study the question of the algebraic difference independence of the Euler gamma function Γ and the functions in a certain class F , which contains those Dirichlet series as L-functions in the extended Selberg class S ♯ and some periodic functions. The main results in this paper are the difference analogues of the corresponding results from Lü (J Math Anal Appl 462(2):1195–1204, 2018) that showed that the Euler gamma function Γ and the functions in F can not satisfy a class of algebraic differential equations with meromorphic coefficients ϕ of Nevanlinna's characteristics satisfying T (r , ϕ) = o (r) , as r → ∞. Examples are provided to show that the main results in this paper, in a sense, are best possible. [ABSTRACT FROM AUTHOR]

Recently a function was constructed that satisfies all known properties of a tree-level scattering of four massless scalars via the exchange of an infinite tower of particles with masses given by the non-trivial zeroes of the Riemann zeta function. A key ingredient in the construction is an even entire function whose only zeroes coincide with the non-trivial zeroes of the Riemann zeta function. In this paper we show that exactly the same conclusions can be drawn for an infinite class of even entire functions with only zeroes on the real line. This shows that the previous result does not seem to be connected to specific properties of the Riemann zeta function, but it applies more generally. As an application, we show that exactly the same conclusions can be drawn for L-functions other than the Riemann zeta function. [ABSTRACT FROM AUTHOR]

By establishing recurrence relations and then determining boundary values, we examine four classes of definite integrals of over higher powers of , , and in denominators. They are explicitly evaluated in terms of the logarithm function, the Riemann zeta function and its variants, such as Dirichlet beta function and Legendre's chi-function. [ABSTRACT FROM AUTHOR]

In this paper, the authors established a sharp version of the difference analogue of the celebrated Hölder's theorem concerning the differential independence of the Euler gamma function Γ. More precisely, if P is a polynomial of n + 1 variables in ℂ[X, Y0, ⋯, Yn−1] such that P (s , Γ (s + a 0) , ⋯ , Γ (s + a n − 1)) ≡ 0 for some (a0, ⋯, an−1) ∈ ℂn and ai − aj ∉ ℤ for any 0 ≤ i ≤ j ≤ n − 1, then they have P ≡ 0. . Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by Hölder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006. [ABSTRACT FROM AUTHOR]

In this paper, we study Multiple slit diffraction and n- array linear antennae in negative refractive index material. We show that the evanescent wave plays a vital role in the near-field term. The evanescent wave grows significantly, unlike in conventional materials, and satisfies a novel kind of convergence known as Cesàro convergence. We calculate the intensity of multiple slits and the antenna's amplification factor (AF) in terms of the Riemann zeta function. We further demonstrate that the Riemann zeta function gives rise to additional nulls. We deduce that all the diffraction scenarios in which the traveling wave satisfies the geometric series in the medium of the positive refractive index will enhance the evanescent wave, which satisfies Cesàro convergence in the medium of the negative refractive index. [ABSTRACT FROM AUTHOR]

The distribution of Riemann zeros may represent a spectrum of a certain point distribution. Potentially suitable distributions were synthesized by summing complex exponentials over Riemann zeros. If the magnitudes of all harmonics are equal to each other, the resulting spectrum has peaks at the logarithms of primes and prime powers in accordance with the von Mangoldt function. If the magnitudes are set inversely proportional to the derivative of the Riemann zeta function at zeros, the spectrum has peaks at the logarithms of primes and products of distinct primes following the Möbius function. Combining trigonometric series over Riemann zeros with the Möbius function, we obtained the spectrum that consists of equal intensity peaks at the logarithms of primes only. [ABSTRACT FROM AUTHOR]

We study some of the analytic properties of a Dirichlet series defined by the sequence of the Tribonacci numbers, such as its analytic continuation to a meromorphic function of the whole plane and some of its special values. These results are generalisations of earlier properties already established for the Fibonacci Dirichlet series and other Dirichlet series defined by degree 2 recurrence sequences. [ABSTRACT FROM AUTHOR]

We consider an inequality concerning absolute values of the Riemann zeta function at places symmetric with respect to the critical line. The study concludes the investigations started by Spira and complements Trudgian's results. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions L k (a , α ; p , q) , which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result. [ABSTRACT FROM AUTHOR]

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other L-functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp's abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group. [ABSTRACT FROM AUTHOR]

Kirchhoff showed that the number of spanning trees of a graph is the spectral determinant of the combinatorial Laplacian divided by the number of vertices; we reframe this result in the quantum graph setting. We prove that the spectral determinant of the Laplace operator on a finite connected metric graph with standard (Neumann–Kirchhoff) vertex conditions determines the number of spanning trees when the lengths of the edges of the metric graph are sufficiently close together. To obtain this result, we analyze an equilateral quantum graph whose spectrum is closely related to spectra of discrete graph operators and then use the continuity of the spectral determinant under perturbations of the edge lengths. [ABSTRACT FROM AUTHOR]

In his lost notebook, Ramanujan noted down many elegant identities involving divisor functions and the modified K-Bessel function, and some of them are connected with the Fourier series expansion of the non-holomorphic Eisenstein series. Recently, Cohen established interesting generalizations of some of the identities of Ramanujan. In this paper, we study Ramanujan and Cohen-type identities associated to a generalized divisor function and the modified K-Bessel function. In the process, we extend a result of Chandrasekharan and Narasimhan and some identities of Cohen. Furthermore, we obtain a new identity for odd zeta values that can be thought of as a Bessel function analogue of Ramanujan's famous formula for odd zeta values. [ABSTRACT FROM AUTHOR]

We study the a-points of partial sums of the Riemann zeta function for any a ∈ ℂ. Our main goal is to understand where in the complex plane do the a-values of partial sums lie and how many a-values are there up to any given height in that region. [ABSTRACT FROM AUTHOR]

In this article we prove meromorphic continuation of weighted zeta functions Z f in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016). We obtain a residue formula proving equality between residues of Z f and invariant Ruelle distributions. We combine this equality with results of Guillarmou et al. (Ann Henri Lebesgue 4:81–119, 2021) in order to relate the residues to Patterson–Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems. [ABSTRACT FROM AUTHOR]

For any real β 0 ∈ [ 1 2 , 1) , let GRH [ β 0 ] be the assertion that for every Dirichlet character χ and all zeros ρ = β + i γ of L (s , χ) , one has β ⩽ β 0 (in particular, GRH [ 1 2 ] is the Generalized Riemann Hypothesis). In this paper, we show that the validity of GRH [ 9 10 ] depends only on certain distributional properties of the zeros of the Riemann zeta function ζ (s) . No conditions are imposed on the zeros of nonprincipal Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

In a recent work of the author, a de Branges space was constructed, as well as an operator on it whose spectrum coincides with the set of nontrivial zeros of the Riemann zeta function after a rotation of the complex plane. Also the canonical system related to the de Branges space in question was constructed. In this paper, a natural factorization is exhibited for the unitary operator that realizes the unitary correspondence between the Hilbert space of the canonical system and the de Branges space, as the superposition of four unitary operators. Bibliography: 1 title. [ABSTRACT FROM AUTHOR]

The Hankel matrices considered in this article arose in one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that, in the case of the Riemann zeta function, the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when the zeta function is replaced by a function having a single zero. For this case, we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors. [ABSTRACT FROM AUTHOR]

The paper studies the zeta function of the monoid generated by prime numbers of the form 3n + 2. The main monoid and the main set are defined. For the corresponding zeta functions, explicit finite formulas are found that give their analytic continuations to the entire complex plane, except for a countable set of poles. Inverse series for these zeta functions are found, and functional equations are derived. Three new types of monoids of natural numbers with a unique prime factorization are defined, namely, monoids of powers, Euler monoids modulo q, and unit monoids modulo q. Expressions for their zeta functions in terms of the Euler product are given. The Davenport–Heilbronn effect for zeta functions of monoids of natural numbers is discussed, which means the appearance of zeros of the zeta functions for terms obtained by decomposition into residue classes to some modulus. For monoids with an exponential sequence of primes, the barrier series hypothesis is proved and it is shown that the holomorphic domain of the zeta function of such a monoid is the complex half-plane to the right of the imaginary axis. In conclusion, topical problems with zeta functions of monoids of natural numbers that require further investigation are considered. [ABSTRACT FROM AUTHOR]

We discuss two hybrid mean value problems by using the properties of Dirichlet L-functions. The first one is involving a new sum analogue of Dedekind sum and Hurwitz zeta-function, while the second is about Hardy sum S5(h, p) and general Kloosterman sum K (n , r , λ ; p) , attached a Dirichlet character λ modulo p. Consequently, we give several exact computational formulas, which generalize some conclusions of this context. [ABSTRACT FROM AUTHOR]

We present generalizations of some results on the integer parts of the reciprocal remainders of the zeta function ζ (s) with s = 2 and s = 3 , and a very short and elegant proof of a recent result on the integer parts of the reciprocal remainders of the series ζ (3) . We also give some historical and theoretical remarks to problems of this type, conduct some analyses, and make some connections with the theory of linear difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm. [ABSTRACT FROM AUTHOR]

We present generalizations of some results on the integer parts of the reciprocal remainders of the zeta function ζ (s) with s = 2 and s = 3 , and a very short and elegant proof of a recent result on the integer parts of the reciprocal remainders of the series ζ (3) . We also give some historical and theoretical remarks to problems of this type, conduct some analyses, and make some connections with the theory of linear difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters (n , ε , q) is given in terms of Chebyshev polynomials. According to the parity of n and the sign ε , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or ε = - 1 using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture. [ABSTRACT FROM AUTHOR]

We proof a Garrett–Böcherer decomposition of a vector-valued Siegel Eisenstein series E l , 0 2 of genus 2 transforming with the Weil representation of Sp 2 (Z) on the group ring C [ (L ′ / L) 2 ] . We show that the standard zeta function associated to a vector-valued common eigenform f for the Weil representation can be meromorphically continued to the whole s-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of f and E l , 0 2 . [ABSTRACT FROM AUTHOR]

An interpolation function of the multiple harmonic sum is defined. Its basic properties, asymptotic behaviors and derivatives, which play fundamental roles in the study of the multiple polylogarithm and multiple zeta values, are investigated. [ABSTRACT FROM AUTHOR]