Multiple Sine Functions Research Articles

Quantum Riemann-Hilbert problems for the resolved conifold

We study the quantum Riemann-Hilbert problems determined by the refined Donaldson-Thomas theory on the resolved conifold. Using the solutions to classical Riemann-Hilbert problems in [6] we give explicit solutions in terms of multiple sine functions with unequal parameters. The new feature of the solutions is that the valid region of the quantum parameter q12=exp⁡(πiτ) varies on the space of stability conditions and BPS t-plane. Comparing the solutions with the partition function of refined Chern-Simons theory and invoking large N string duality, we find that the solution contains the non-perturbative completion of the refined topological string on the resolved conifold. Therefore solving the quantum Riemann-Hilbert problems provides a possible non-perturbative definition for the Donaldson-Thomas theory.

On refined Chern-Simons/topological string duality for classical gauge groups

We present the partition function of the refined Chern-Simons theory on S3 with arbitrary A, B, C, D gauge algebra in terms of multiple sine functions. For B and C cases this representation is novel. It allows us to conjecture duality to some refined and orientifolded versions of the topological string on the resolved conifold, and carry out the detailed identification of different contributions. The free energies for D and C algebras possess the usual halved contribution from the A theory, i.e. orientable surfaces, and contributions of non-orientable surfaces with one cross-cup, with opposite signs, similar as for the non-refined theories. However, in the refined case, both theories possess in addition a non-zero contribution of orientable surfaces with two cross-cups. In particular, we observe a trebling of the Kähler parameter, in the sense of a refinement and world-sheet (i.e. the number of cross-cups) dependent quantum shift. For B algebra the contribution of Klein bottles is zero, as is the case in the non-refined theory, and the one-cross-cup terms differ from the D and C cases. For the (refined) constant maps terms of these theories we suggest a modular-invariant representation, which leads to natural topological string interpretation. We also calculate some non-perturbative corrections.

Absolute multiple sine functions

In this paper we formulate a unified theory of multiple sine functions by using a view point of absolute zeta functions and absolute automorphic forms.

Multiple elliptic gamma functions associated to cones

We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of SLr(Z)-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.

Dualities for absolute zeta functions and multiple gamma functions

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.

Formal group laws for multiple sine functions and applications

We investigate addition relations for multiple sine functions from the view point of formal group laws. We find that the functions which appear in the coefficients are related to classical Eisenstein serires. As application we obtain a limit formula for automorphic forms.

Determinants and conformal anomalies of GJMS operators on spheres

The conformal anomalies and functional determinants of the Branson-GJMS operators, P2k, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k ⩽ d/2 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non-existence of ‘super-critical’ operators, 2k > d for even d. Significant use is made of the Barnes zeta function. The results are given in terms of ratios of determinants of operators on a (d + 1)-dimensional bulk dual sphere. For odd dimensions, the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations, yielding an integral over a Plancherel measure. The N–D determinant ratio is also found explicitly for even dimensions. Ehrhart polynomials are encountered.

Generalized log sine integrals and the Mordell-Tornheim zeta values

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.

Extremal values of multiple gamma and sine functions

We study the extremal values of multiple gamma and sine functions in the fundamental intervals. We show the number and locations of the extremal points, and prove that all the local maximum and minimum values are greater and less than one, respectively.

ITERATED EULER'S INTEGRALS AND NORMALIZED MULTIPLE SINE FUNCTIONS

We introduce iterated Euler's integrals, and we give expressions using zeta functions. Moreover, we prove that normalized multiple sine functions are expressed via iterated Euler's integrals. Then, we show basic properties of multiple sine functions from these results containing Kummer type formula.

Weierstrass product representations of multiple gamma and sine functions

It is well known that the Weierstrass product representation of the Barnes multiple gamma function Γr(z) can be calculated concretely. However, there has been no study on its explicit formulation. In this paper, its simple formulation is achieved. It is applicable to the Weierstrass product representation of the Vignéras multiple gamma function also. Moreover, the Weierstrass product representation of the Kurokawa multiple sine function Sr(z) is also formulated explicitly.

Algebraic relations between special values of multiple sine functions

Algebraic relations between special values of multiple sine functions

Central values of generalized multiple sine functions

We give a nontrivial estimate of the central value of the generalized multiple sine function by a new integral expression of its logarithm. Moreover, when all the periods coincide with 1, we show relations between the central values of the generalized multiple sine functions and the special values of the Riemann zeta function.

BEHAVIOR OF SOME GENERALIZED MULTIPLE SINE FUNCTIONS

Our aim is to investigate the behavior of generalized multiple sine functions with general period parameters in the fundamental domain. For that, we need to calculate the number of their extremal values. By estimating their special values, we determine it in some cases including the quintuple sine function. As a consequence, we sketch their graphs.

PERIOD DEFORMATIONS OF MULTIPLE HURWITZ ZETA FUNCTIONS

We study continuous period deformations of the multiple Hurwitz zeta functions and their derivatives. Moreover we investigate period deformations of the generalized multiple gamma and sine functions and give applications.

Mahler measures and computations with regulators

In this work we apply the techniques that were developed in [M.N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Y. Gon, H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (5) (2004) 425–442].

Zeta functions and normalized multiple sine functions

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.

Shintani’s prehomogeneous zeta functions and multiple sine functions

We show that the derivative at 0 of Shintani’s prehomogeneous zeta function for the space of symmetric matrices is expressed via special values of multiple sine functions.

EULER PRODUCT EXPRESSION OF TRIPLE ZETA FUNCTIONS

We construct multiple zeta functions considered as absolute tensor products of usual zeta functions. We establish Euler product expressions for triple zeta functions [Formula: see text] with p, q, r distinct primes, via multiple sine functions by using the signatured Poisson summation formula. We also establish Euler product expressions for triple zeta functions [Formula: see text] with a prime p, via the theory of multiple sine functions.

Differential Algebraicity of Multiple Sine Functions

The differential algebraicity of the multiple sine functions \(S_{r} (x, \underline{\omega})\) is investigated. The goal of the paper is to show the differential algebraicity of \(S_{r} (x, \underline{\omega})\) when the period \(\underline{\omega})\) is “rational” by using the classical theory due to Ostrowski.