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The zeta function is like a milestone in analytic number theory. Considering Euler's definition of the zeta function in this article, we will find some recursion at even integral points. We will also discuss the Dirichlet eta function, even Bernoulli numbers, and the gamma–zeta relation. Toward the end, we will define something known as the odd zeta function and find a recurrence for the same. [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]

In this paper the product of classical Riemann zeta function ζ (s) and fractional hypergeometric zeta functions ζ a (s) is examined. As a result of this product a new integral representation of the square of the Riemann zeta function is obtained. We use this integral representation to show that some special Mellin transforms which are naturally related to the fractional hypergeometric zeta functions have zero free regions in the specified right half of the complex plane. Finally we show that the fractional hypergeometric zeta functions ζ a (s) are zero free for some specified values of a. [ABSTRACT FROM AUTHOR]

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks. Afterwards we deal with higher-dimensional quantum walks. For comparison with the case of the quantum walk, we also treat the case of higher-dimensional random walks. Our results bridge between the Mahler measure and the zeta function via quantum walks for the first time. [ABSTRACT FROM AUTHOR]

As the generalization of the digraph, the directed hypergraph (dihypergraph), denoted by H , is considered in this paper. First, we introduce the notions of the zeta function of H and dihypergraph coverings over H . Next, several expressions for the zeta function of H are given, and all dihypergraph coverings over H are generated by permutation voltage assignments on the incidence bipartite digraph or the edge-colored digraph of H . Final, we derive two explicit decomposition formulae for the zeta function of any dihypergraph covering H ¯ over H , which turn out that the zeta function of H divides the zeta function of H ¯ . [ABSTRACT FROM AUTHOR]

Recently, the Ihara zeta function for the finite graph has been extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno–Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno–Sato theorem "Grover/Zeta Correspondence" here. [ABSTRACT FROM AUTHOR]

It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier–Laplace transform of a unique exponentially finite measure on R . In this paper we apply this complex integral representation to specific families of special functions, including the Γ , ζ and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the ζ function this process leads to a new proof of an integral representation on the critical strip. [ABSTRACT FROM AUTHOR]

We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes. [ABSTRACT FROM AUTHOR]

We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation between the spectrum of the Laplacian on a discrete graph and that of the Laplacian on an equilateral metric graph. As a by-product, we determine how the multiplicity of eigenvalues of the quantum graph, that are also in the spectrum of the graph with Dirichlet conditions at the vertices, depends on the graph geometry. Finally we apply the result to calculate the vacuum energy and spectral determinant of a complete bipartite graph and compare our results with those for a star graph, a graph in which all vertices are connected to a central vertex by a single edge. [ABSTRACT FROM AUTHOR]

We establish a joint universality theorem for pairs of functions in the Selberg class under certain conditions. This theorem generalizes and unifies several previous results, which were shown individually. We also give further examples of pairs of jointly universal L-functions, and actually extend the known universality theorem for the symmetric power L-function $${L(s, \mathrm{sym}^m f)}$$ associated to a holomorphic Hecke eigen cusp form f for $${\mathrm{SL}_{2} (\mathbb{Z})}$$ with $${1 \le m \le 4}$$ . [ABSTRACT FROM AUTHOR]

We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the semigroup unsettles the ergodic properties of the action. [ABSTRACT FROM AUTHOR]

The problem of finding the number and the most likely shape of solutions of the system $$\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N$$ , where λ , M, N > 0 and N is an integer, as M, N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ is of the order of λ , where d, the number of degrees of freedom, is less than two. [ABSTRACT FROM AUTHOR]

We give a proof of a formula for the Bartholdi zeta function of a digraph by the method of Stark and Terras (Adv. Math. 121:124-165, ). [ABSTRACT FROM AUTHOR]

Let S be a bielliptic surface over a finite field, and let an elliptic curve B be the Albanese variety of S; then the zeta function of the surface S is equal to the zeta function of the direct product P × B. Therefore, the classification problem for the zeta functions of bielliptic surfaces is reduced to the existence problem for surfaces of a given type with a given Albanese curve. In the present paper, we complete this classification initiated in [1]. [ABSTRACT FROM AUTHOR]

We extend a balanced signed graph to a digraph, and present a necessary and sufficient condition for a signed digraph to be balanced. Moreover, we give another necessary and sufficient condition for a signed digraph $$(D,w)$$ to be balanced by using zeta functions of $$D$$ . As an application, we discuss the structure of balanced coverings of signed digraphs under consideration of coverings of strongly connected digraphs. [ABSTRACT FROM AUTHOR]

In this article we present results on the Shannon information (SI) contained in upper (lower) record values and associated record times in a sequence of i.i.d continuous variables. We then establish an interesting relationship between the SI content of a random sample of fixed size, and the SI in the data consisting of sequential maxima. We also consider the information contained in the record data from an inverse sampling plan (ISP). [ABSTRACT FROM AUTHOR]

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c, converge rapidly to zero as k→∞. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc ( at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d. These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula. [ABSTRACT FROM AUTHOR]

In the framework of the zeta-regularization approach, we consider the polarization of the scalar field vacuum with nontrivial boundary conditions originating from electrodynamics in the presence of a conducting infinitely thin boundary layer. Boundary conditions of the first type correspond to the case where the field is continuous on the boundary while its derivative has a jump proportional to the boundary value of the field. Boundary conditions of the second type correspond to the case where the field derivative is continuous on the boundary but the field itself has a jump proportional to the field derivative on the boundary. We explicitly obtain the zeta function of the scalar field Laplace operator with the above boundary conditions and calculate all the heat kernel coefficients. We obtain an expression for the energy of the scalar field vacuum fluctuations. [ABSTRACT FROM AUTHOR]

We prove that an appropriately modified Mellin transform of the Hardy function Z( x) Is en entire function. The proof is based on the fact that the function (2 − 1) ζ( s) is integer. [ABSTRACT FROM AUTHOR]

We prove the 2-out-of-3 property for the rationality of the motivic zeta function in distinguished triangles in Voevodsky’s category [Figure not available: see fulltext.]. As an application, we show the rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in [Figure not available: see fulltext.]. Together with a result due to P. O’sullivan, this also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational. [ABSTRACT FROM AUTHOR]

We consider lower bounds for the Euler—Kronecker constant in the case of number fields and upper and lower bounds in the case of algebraic manifolds over a finite field. [ABSTRACT FROM AUTHOR]

We prove new relations for multiple zeta values. In particular, they imply Vasil’ev’s equality and a formula for the summation of multiple zeta values of fixed weight with a constraint on the first coordinate. [ABSTRACT FROM AUTHOR]

Let S be a bielliptic surface over a finite field, and let the elliptic curve B be the image of the Albanese mapping S → B. In this case, the zeta function of the surface is equal to the zeta function of the direct product ℙ1 × B. A classification of the possible zeta functions of bielliptic surfaces is also presented in the paper. [ABSTRACT FROM AUTHOR]

A Voronin-type discrete universality theorem for the L-functions of new forms is proved. [ABSTRACT FROM AUTHOR]

Let p1:K?H, p2:H?Gand p2p1:K?Gbe three finite regular coverings of graphs, and let s be a representation of the covering transformation group of p1. We show that the (Bartholdi type)L-function ofGassociated to the representation of the covering transformation group of p2p1 induced from s is equal to that ofHassociated to s by means of ordinary voltage assignments. [ABSTRACT FROM AUTHOR]

We present a polynomial-time algorithm for computing the zeta function of a smooth projective hypersurface of degree d over a finite field of characteristic p, under the assumption that p is a suitably small odd prime and does not divide d. This improves significantly upon an earlier algorithm of the author and Wan which is only polynomial-time when the dimension is fixed. [ABSTRACT FROM AUTHOR]

We present a practical polynomial-time algorithm for computing the zeta function of a Kummer curve over a finite field of small characteristic. Such algorithms have recently been obtained using a method of Kedlaya based upon Monsky–Washnitzer cohomology, and are of interest in cryptography. We take a different approach. The problem is reduced to that of computing the L-function of a multiplicative character sum. This latter task is achieved via a cohomological formula based upon the work of Dwork and Reich. We show, however, that our method and that of Kedlaya are very closely related. [ABSTRACT FROM AUTHOR]

Shows a proof of a series with simple zeros of zeta function. Riemann conjecture; Riemann hypothesis; Holder's inequality; Proof of the theorem.

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the quantitative equidistribution of the shape of cubic fields when the fields are ordered by discriminant. [ABSTRACT FROM AUTHOR]