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Nobushige Kurokawa - One of the best experts on this subject based on the ideXlab platform.
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Milnor-Selberg zeta functions and zeta regularizations
Journal of Geometry and Physics, 2013Co-Authors: Nobushige Kurokawa, Masato Wakayama, Yoshinori YamasakiAbstract:Abstract By a similar idea for the construction of Milnor’s gamma functions, we introduce “higher depth determinants” of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor–Selberg zeta function. It is shown that the Milnor–Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.
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Zeta functions and normalized multiple sine functions
Kodai Mathematical Journal, 2005Co-Authors: Shinya Koyama, Nobushige KurokawaAbstract:By using normalized multiple sine functions we show expressions for special values of zeta functions and L-functions containing ζ(3), ζ(5), etc. Our result reveals the importance of division values of normalized multiple sine functions. Properties of multiple Hurwitz zeta functions are crucial for the proof. Mathematics Subject Classification 2000: 11M06
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Extensions of Zeta Functions – Examples and a Study of the Double Sine Functions
Acta Applicandae Mathematica, 2005Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al. , Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys. , 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.
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Extensions of Zeta Functions - Examples and a Study of the Double Sine Functions
Acta Applicandae Mathematicae, 2005Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.
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multiple zeta functions the double sine function and the signed double poisson summation formula
Compositio Mathematica, 2004Co-Authors: Shinya Koyama, Nobushige KurokawaAbstract:We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case $\zeta(s,\mathbf{F}_p)\otimes\zeta(s,\mathbf{F}_q)$ by using the signed double Poisson summation formula and the theory of the double sine function.
Masato Wakayama - One of the best experts on this subject based on the ideXlab platform.
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Milnor-Selberg zeta functions and zeta regularizations
Journal of Geometry and Physics, 2013Co-Authors: Nobushige Kurokawa, Masato Wakayama, Yoshinori YamasakiAbstract:Abstract By a similar idea for the construction of Milnor’s gamma functions, we introduce “higher depth determinants” of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor–Selberg zeta function. It is shown that the Milnor–Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.
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Extensions of Zeta Functions - Examples and a Study of the Double Sine Functions
Acta Applicandae Mathematicae, 2005Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.
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Extensions of Zeta Functions – Examples and a Study of the Double Sine Functions
Acta Applicandae Mathematica, 2005Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al. , Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys. , 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.
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Higher Selberg Zeta Functions
Communications in Mathematical Physics, 2004Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z _Γ( s ), especially to obtain the functional equation. We also describe the gamma factor of z _Γ( s ) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.
Antanas Laurincikas - One of the best experts on this subject based on the ideXlab platform.
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Joint universality of the Riemann zeta-function and Lerch Zeta-Functions
Nonlinear Analysis: Modelling and Control, 2013Co-Authors: Antanas Laurincikas, Renata MacaitienėAbstract:In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch Zeta-Functions with parameters algebraically independent over the field of rational numbers.
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On joint universality of the Riemann zeta-function and Hurwitz Zeta-Functions☆
Journal of Number Theory, 2012Co-Authors: Antanas LaurincikasAbstract:Abstract We construct classes of composite functions of the Riemann zeta-function and Hurwitz zeta function with transcendental parameter which are universal in the sense that their shifts uniformly on compact subsets of some region approximate any analytic function. For example, the functions c 1 ζ ( s ) + c 2 ζ ( s , α ) , c 1 , c 2 ∈ C ∖ { 0 } , e ζ ( s ) + ζ ( s , α ) and sin ( ζ ( s ) + ζ ( s , α ) ) are universal.
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The Universality of Zeta-Functions
Acta Applicandae Mathematicae, 2003Co-Authors: Antanas LaurincikasAbstract: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.
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The Lerch Zeta-function
2003Co-Authors: Antanas Laurincikas, Ramūnas GarunkštisAbstract:The Lerch zeta-function is the first monograph on this topic, which is a generalization of the classic Riemann, and Hurwitz Zeta-Functions. Although analytic results have been presented previously in various monographs on Zeta-Functions, this is the first book containing both analytic and probability theory of Lerch Zeta-Functions. The book starts with classical analytical theory (Euler gamma-functions, functional equation, mean square). The majority of the presented results are new: on approximate functional equations and its applications and on zero distribution (zero-free regions, number of nontrivial zeros etc). Special attention is given to limit theorems in the sense of the weak convergence of probability measures for the Lerch zeta-function. From limit theorems in the space of analytic functions the universitality and functional independence is derived. In this respect the book continues the research of the first author presented in the monograph Limit Theorems for the Riemann zeta-function. This book will be useful to researchers and graduate students working in analytic and probabilistic number theory, and can also be used as a textbook for postgraduate students
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limit theorems for the riemann zeta function
1995Co-Authors: Antanas LaurincikasAbstract:Preface. 1. Elements of the probability theory. 2. Dirichlet series and Dirichlet polynomials. 3. Limit theorems for the modulus of the Riemann Zeta-function. 4. Limit theorems for the Riemann Zeta-function on the complex plane. 5. Limit theorems for the Riemann Zeta-function in the space of analytic functions. 6. Universality theorem for the Riemann Zeta-function. 7. Limit theorem for the Riemann Zeta-function in the space of continuous functions. 8. Limit theorems for Dirichlet L-functions. 9. Limit theorem for the Dirichlet series with multiplicative coefficients. References. Notation. Subject index.
Hirofumi Tsumura - One of the best experts on this subject based on the ideXlab platform.
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Certain convolution formulas for multiple series
The Ramanujan Journal, 2013Co-Authors: Hirofumi TsumuraAbstract: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.
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Multiple Zeta-Functions associated with linear recurrence sequences and the vectorial sum formula.
Canadian Journal of Mathematics, 2011Co-Authors: Driss Essouabri, Kohji Matsumoto, Hirofumi TsumuraAbstract:We prove the holomorphic continuation of certain multi-variable multiple Zeta-Functions whose coefficients satisfy a suitable recurrence condition. Actually we introduce more general vectorial Zeta-Functions, and prove their holomorphic continuation. Moreover we show a vectorial sum formula among those vectorial Zeta-Functions, from which some generalizations of the classical sum formula can be deduced.
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on witten multiple zeta functions associated with semisimple lie algebras ii
Journal of The Mathematical Society of Japan, 2010Co-Authors: Yasushi Komori, Kohji Matsumoto, Hirofumi TsumuraAbstract:We prove certain general forms of functional relations among Witten multiple Zeta-Functions in several variables (or Zeta-Functions of root systems). The structural background of these functional relations is given by the symmetry with respect to Weyl groups. From these relations, we can deduce explicit expressions of values of Witten Zeta-Functions at positive even integers, which are written in terms of generalized Bernoulli numbers of root systems. Furthermore, we introduce generating functions of Bernoulli numbers of root systems, using which we can give an algorithm of calculating Bernoulli numbers of root systems.
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on functional relations between the mordell tornheim double zeta functions and the riemann zeta function
Mathematical Proceedings of the Cambridge Philosophical Society, 2007Co-Authors: Hirofumi TsumuraAbstract:In this paper, we give certain analytic functional relations between the Mordell?Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell?Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.
Junesang Choi - One of the best experts on this subject based on the ideXlab platform.
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certain relationships among polygamma functions riemann zeta function and generalized zeta function
Journal of Inequalities and Applications, 2013Co-Authors: Junesang Choi, Chaoping ChenAbstract:Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ (s) and the Hurwitz zeta function ζ (s, a) have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen's method. We also present a double inequality approximating ζ (2r + 1) by a more rapidly convergent series. MSC: Primary 11M06; 33B15; secondary 40A05; 26D07
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Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function
Journal of Inequalities and Applications, 2013Co-Authors: Junesang Choi, Chaoping ChenAbstract:Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ ( s ) Open image in new window and the Hurwitz zeta function ζ ( s , a ) Open image in new window have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. We also present a double inequality approximating ζ ( 2 r + 1 ) Open image in new window by a more rapidly convergent series.
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The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function
Taiwanese Journal of Mathematics, 2011Co-Authors: Junesang Choi, Hari M. SrivastavaAbstract:Almost eleven decades ago, Barnes introduced and made a \linebreak systematic investigation on the multiple Gamma functions $\Gamma_n$. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the $n$-dimensional unit sphere ${\bf S}^n$ by using the multiple Hurwitz zeta functions $\zeta_n(s,a)$. In this paper, we first aim at presenting a generalized Hurwitz formula for $\zeta_n(s,a)$ together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function $\eta_n(s,a)$ in two different ways. As a by-product of our second investigation, a relationship between $\;\eta_n(-\ell,a)$ $\;(\ell \in \mathbb{N}_0)$ and the generalized Euler polynomials $E_\ell^{(n)}(n-a)$ is also presented.
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series involving the zeta function and multiple gamma functions
Applied Mathematics and Computation, 2004Co-Authors: Junesang Choi, H M SrivastavaAbstract:The theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, was applied in several earlier works in order to evaluate some families of series involving the Riemann Zeta function as well as to compute the determinants of the Laplacians. Here, in the present paper, the authors address the converse problem and apply various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions and then using these series to compute the determinants of the Laplacians on the n-dimensional unit sphere S^n (n=5,6,7) explicitly.
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certain classes of series associated with the zeta function and multiple gamma functions
Journal of Computational and Applied Mathematics, 2000Co-Authors: Junesang Choi, H M SrivastavaAbstract:Abstract The authors apply the theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, in order to evaluate some families of series involving the Riemann Zeta function. By introducing a certain mathematical constant, they also systematically evaluate this constant and some definite integrals of the triple Gamma function. Various classes of series associated with the Zeta function are expressed in closed forms. Many of these results are also used here to compute the determinant of the Laplacian on the four-dimensional unit sphere S 4 explicitly.