Asymptotic expansions for a class of zeta-functions

In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy. In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being incorporated. In Sect. 2, we treat the Taylor series expansion of the Lipschitz–Lerch transcendent in the perturbation variable. In the proofs, we make an extensive use of the beta-transform (used to be called the Mellin–Barnes formula).     

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

We give explicit approximate functional equations for various classes of Dirichlet series, such as Lerch zeta function or Dirichlet L-series.Dedicated to Jean-Louis Nicolas on the occasion of his sixtieth birthday2000 Mathematics Subject Classification: Primary—11M06, 11M35, 11Y60     

Growth of the Lerch zeta-function

It is believed that the Lindelöf hypothesis is also true for the Lerch zeta-function. Here we present results supporting this conjecture. We first consider the growth of the Lerch zeta-function assuming the generalized Lindelöf hypothesis for Dirichlet L-functions. We next prove that Huxleys exponent 32/205 in the Lindelöf hypothesis for the Riemann zeta-function holds also for the Lerch zeta-function.__________Partially supported by a grant from the Lithuanian State Science and Studies Foundation.Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 45–56, January–March, 2005.     

Twists of Lerch Zeta-Functions

We study twists of the Lerch zeta-functions with shifted Dirichlet characters. This generalizes Dirichlet L-functions. These twists turn out to be meromorphic with at most one simple pole and satisfy a functional equation (similar to the functional equation of the Lerch zeta-function) if the Dirichlet character is primitive. We also investigate the distributions of their zeros. In particular, we disprove the analogue of Riemann's hypothesis for some twists.     

Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.     

The existence and the non-existence of joint t-universality for Lerch zeta functions

In this paper, we show the following theorems. Suppose 0<al<1 are algebraically independent numbers and 0<λl?1 for 1?l?m. Then we have the joint t-universality for Lerch zeta functions L(λl,al,s) for 1?l?m. Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions.     

The universality for linear combinations of Lerch zeta functions and the Tornheim�CHurwitz type of double zeta functions

In this paper, we consider the universality for linear combinations of Lerch zeta functions. J. Kaczorowski, A. Laurin?ikas and J. Steuding treated universal Dirichlet series with the case that the compact sets ${\mathcal{K}_l}$ are disjoint. But we consider the both cases that the compact subset ${\mathcal{K}_l}$ is disjoint and not disjoint. Next, we will show the non-trivial zeros of the Tornheim?CHurwitz type of double zeta functions in the region of absolute convergence. Moreover we show the universality for the Tornheim?CHurwitz type of double zeta function.     

On the apostol-bernoulli polynomials

In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.     

Tauberian conditions for w-almost convergent double sequences

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ L^p(T²), where 1 < p < ∞.      

Power series approximations for two-class generalized processor sharing systems

We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1−β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z ₁,z ₂) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z ₁,z ₂) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.     

On generalized statistical convergence of double sequences via ideals

The concept of statistical convergence is one of the most active area of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper we generalize the notions of statistical convergence, (λ, μ)-statistical convergence, (V, λ, μ) summability and (C, 1, 1) summability for a double sequence x = (x _jk ) via ideals. We also establish the relation between our new methods.     

Series Acceleration Formulas for Dirichlet Series with Periodic Coefficients

Series acceleration formulas are obtained for Dirichlet series with periodic coefficients. Special cases include Ramanujan's formula for the values of the Riemann zeta function at the odd positive integers exceeding two, and related formulas for values of Dirichlet L-series and the Lerch zeta function.     

Extension of a theorem of Fejér to double Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π^-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:

A hypergeometric approach,via linear forms involving logarithms,to criteria for irrationality of Euler’s constant

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.      

Generalized Zeta Regularizations,Quantum Class Number Formulas,and Appell's <Emphasis Type="Italic">O</Emphasis>-Functions

We introduce a generalized notion of the zeta regularization. As applications we show the quantum analogue of the Lerch formula and of the Dirichlet class number formulas. We also give some further examples related to Appell's O-functions. 2000 Mathematics Subject Classification: Primary—11M36 Work in part supported by Grant-in Aid for Scientific Research (B) No. 11440010, and by Grant-in Aid for Exploratory Research No. 13874004, Japan Society for the Promotion of Science.     

The clone of operations preserving a cycle with loops

We consider all the binary relations on a k-element set which, when viewed as directed graphs, consist of a k-cycle and some loops. If k ≥ 3 and the relation has at least two loops, we show that it is only preserved by essentially unary operations. In all other cases, the relation is preserved by operations that depend on a greater number of variables. Received March 9, 2006; accepted in final form September 14, 2007.     

Some formulas related to Hurwitz–Lerch zeta functions

In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections, various results are always proved by multiplication and inversion formulas.     

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.