Reidemeister and Nielsen zeta-functions

In this paper we formulate a rationality theorem for the Reidemeister and Nielsen zeta-functions modulo a normal subgroup of the fundamental group. We give conditions under which these zeta-functions coincide. We formulate a conjecture aboutentropy for the Reidemeister numbers. We show that the radius of convergence of the Nielsen zeta-function for an orientation-preserving homeomorphism f of a compact surface is an invariant of a three-dimensional manifold, the torus of the map f, and a special flow on it. In special cases we derive a functional equation for the Nielsen zeta-function. We give an example of a transcendental Nielsen zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 164–168, 1988.In conclusion the author expresses thanks to V. B. Piloginaya, V. G. Turaev, Boju Jiang, N. V. Ivanov for stimulating discussions and to D. Fried for sending preprints.     

Certain convolution formulas for multiple series

In this paper, computing a double integral of convolution type in two ways, we give certain formulas for general multiple series. The method is based on that of Kanemitsu–Tanigawa–Yoshimoto in their previous work. As concrete examples, considering multiple zeta-functions of Barnes type and Euler–Zagier type, and Epstein zeta-functions, we give new formulas for multiple series involving these zeta-functions.     

Universality theorems for zeta-functions with periodic coefficients

We obtain Voronin-type universality theorems for some classes of functions of a collection of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.     

The Universality of Zeta-Functions

The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Euler's product as well as zeta-functions without Euler's product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank 3 is proved.     

On Zeta-Functions of Finite Abelian Groups

In the paper, the zeta-functions of finite Abelian groups of rank 2 and 3 are considered. One- and two-dimensional limit theorems for these zeta-functions on the complex plane and in the space of meromorphic functions are obtained.     

Joint discrete universality for periodic zeta-functions

Abstract

In the paper, joint universality theorems for periodic zeta functions with multiplicative coefficients and periodic Hurwitz zeta-functions are proved. The main theorem of [11] is extended, and two new joint universality theorems on the approximation of a collection of analytic functions by discrete shifts of the above zeta-functions are obtained. For this, certain linear independence hypotheses are applied.     

Note on zeros of the derivative of the Selberg zeta-function

Wenzhi Luo studied the distribution of nontrivial zeros of the derivatives of Selberg zeta-functions on cocompact hyperbolic surfaces, and obtained an asymptotic formula for zero density with bounded height. Then he related the distribution of zeros to the multiplicities of Laplacian eigenvalues. Using better bounds for the growth of the Selberg zeta-functions we improve some of the above results. This work was partially supported by a Grant from the Lithuanian Foundation of Studies and Science.     

Applications of hybrid universality to multivariable zeta-functions

In the present paper, we obtain new results on universality as applications of hybrid universality and almost-periodicity in its half-plane of absolute convergence. By using these, we show the universality for Euler products of Igusa type, Euler-Zagier multiple zeta-functions and Tornheim-Hurwitz type of double zeta-functions.     

A NEW METHOD OF PRODUCING FUNCTIONAL RELATIONS AMONG MULTIPLE ZETA-FUNCTIONS

In this paper, we introduce a new method of producing functionalrelations among multiple zeta-functions. This method can beregarded as a kind of multiple analogue of Hardy's one of provingthe functional equation for the Riemann zeta-function. Usingthis method, we give new functional relations for multiple zeta-functions.In particular, substituting positive integers into variablesof them, we obtain known relation formulas for the multiplezeta-values. Furthermore, applying our method to a certain seriesinvolving hyperbolic sine functions, we can obtain certain multipleanalogues of the known results given by Cauchy, Ramanujan, Berndtand so on.     

Bounds for triple zeta-functions

In the present paper we consider the problem of the order of magnitude for the triple zeta-functions of Euler-Zagier type in the region 0≤ ℜsj < 1 (j = 1,2,3). We apply the Euler-Maclaurin summation formula and van der Corput's method of multiple exponential sums to the triple zeta-functions.     

The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I

We consider general multiple zeta-functions of multi-variables, including both Barnes multiple zeta-functions and Euler-Zagier sums as special cases. We prove the meromorphic continuation to the whole space, asymptotic expansions, and upper bound estimates. These results are expected to have applications to some arithmetical L-functions (such as of Hecke and of Shintani). The method is based on the classical Mellin-Barnes integral formula.     

Chaotic secure content-based hidden transmission of biometric templates

The large-scale proliferation of biometric verification systems creates a demand for effective and reliable security and privacy of its data. Like passwords and PIN codes, biometric data is also not secret and if it is compromised, the integrity of the whole verification system could be at high risk. To address these issues, this paper presents a novel chaotic secure content-based hidden transmission scheme of biometric data. Encryption and data hiding techniques are used to improve the security and secrecy of the transmitted templates. Secret keys are generated by the biometric image and used as the parameter value and initial condition of the chaotic map, and each transaction session has different secret keys to protect from the attacks. Two chaotic maps are incorporated for the encryption to resolve the finite word length effect and to improve the system’s resistance against attacks. Encryption is applied on the biometric templates before hiding into the cover/host images to make them secure, and then templates are hidden into the cover image. Experimental results show that the security, performance, and accuracy of the presented scheme are encouraging comparable with other methods found in the current literature.     

The joint universality of Dirichlet L-functions and Lerch zeta-functions

We establish a Voronin-type joint universality theorem on approximating analytic functions by the translations of Dirichlet L-functions and Lerch zeta-functions.     

Shifted poly-Cauchy numbers

Recently, the first author introduced the concept of poly-Cauchy numbers as a generalization of the classical Cauchy numbers and an analogue of poly-Bernoulli numbers. This concept has been generalized in various ways, including poly-Cauchy numbers with a q parameter. In this paper, we give a different kind of generalization called shifted poly-Cauchy numbers and investigate several arithmetical properties. Such numbers can be expressed in terms of original poly-Cauchy numbers. This concept is a kind of analogous ideas to that of Hurwitz zeta-functions compared to Riemann zeta-functions.     

A Limit Theorem for the Arguments of Zeta-Functions of Certain Cusp Forms

In this paper, we obtain a limit theorem for the moduli of the arguments of zeta-functions on the critical line of normalized eigenforms.     

On the argument of zeta-functions of certain cusp forms near the critical line

We prove a limit theorem for the argument of zeta-functions of holomorphic normalized Hecke-eigen cusp forms near the critical line.     

On joint universality of periodic Hurwitz zeta-functions

We prove a joint universality theorem for a collection of periodic Hurwitz zeta-functions with algebraically independent parameters over the field of rational numbers.     

A joint universality theorem for periodic Hurwitz zeta-functions. II

We obtain a joint universality theorem for periodic Hurwitz zeta-functions under weaker hypotheses than those in the previous papers of the first author.     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. IV

We obtain a joint limit theorem in the space of analytic functions for Lerch zeta-functions with algebraic irrational parameter. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-81/09.     

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.