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 generalized Mordell–Tornheim zeta functions

In the theory of complex multiplication, it is important to construct class fields over CM fields. In this paper, we consider explicit K3 surfaces parametrized by Klein’s icosahedral invariants. Via the periods and the Shioda–Inose structures of K3 surfaces, the special values of icosahedral invariants generate class fields over quartic CM fields. Moreover, we give an explicit expression of the canonical model of the Shimura variety for the simplest case via the periods of K3 surfaces.     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday     

Interval functions related to geocze k-area †

The Geocze area theory for dimension two uses the geometrically simplest intervals on which area theory can he developed, These intervals are simple polygonal regions in the plane. For Geocze k-area with k > 2 it is shown in the present paper that the geometrically simplest interval arc not tue obvious generalizations of the- two dimensional case, that is, polyhedral regions in R^k which are topological K-cells. It is shown that the simplest interval is a polyhedral region in R^k whose complement is connected     

关于Hurwitz zeta—函数的四次均值

对ｓ＝σ＋ｉｔ，ζ（ｓ，ａ）是Ｈｕｒｗｉｔｚｚｅｔａ－函数，ζ１（ｓ，ａ）＝ζ（ｓａ）－ａ＾－ｓ，。本文的主要目的是用解析的方法给出了Ｈｕｒｗｉｔｚｚｅｔａ－函数四次均值较强的渐近公式。     

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.     

Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.     

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.     

Some discrete Fourier transform pairs associated with the Lipschitz–Lerch Zeta function

It is shown that there exists a companion formula to Srivastava’s formula for the Lipschitz–Lerch Zeta function [see H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84] and that together these two results form a discrete Fourier transform pair. This Fourier transform pair makes it possible for other (known or new) results involving the values of various Zeta functions at rational arguments to be easily recovered or deduced in a more general context and in a remarkably unified manner.     

Some identities for Appell–Lerch sums and a universal mock theta function

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock \(\theta \)-functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock \(\theta \)-functions denoted by \(g_{_{2}}\) and \(g_{_{3}}\). These functions are normalized level 2 and level 3 Appell–Lerch functions. In the survey paper, the authors list several identities for certain Appell–Lerch functions and refer the proofs to a future paper with this title, listed in their references as [GM3]. The purpose of this paper is to prove these identities. One of the identities removes the \( \theta \) -quotients from Kang’s formulas, which express \(g_{_{2}}\) and \({g}_{{_{3}}}\) in terms of Zwegers’ \(\mu \)-function and \( \theta \)-quotients.     

Hurwitz zeta—函数的两个积分均值公式

设ζ(s,α)为Hurwitzzeta函数.当Re(s)＞1时,定义     

Some Hermite–Hadamard type inequalities for geometrically quasi-convex functions

In the paper, we introduce a new concept ‘geometrically quasi-convex function’ and establish some Hermite–Hadamard type inequalities for functions whose derivatives are of geometric quasi-convexity.     

Zeta functions related to the group SL2(ℤ p )

An explicit formula is given for the number of subgroups of indexp ⁿ in the principle congruence subgroups of SL₂(ℤ _p ) (for odd primesp), and for the zeta function associated with the group. Asymptotically this number iscnp ⁿ , wherec is a constant depending on the congruence subgroup. Also, the zeta function of thei-th congruence subgroup coincides with the partial zeta function of the 3-generated subgroups of thei+1-th congruence subgroup, and for each indexp ⁿ the ratio between 2-generated subgroups and 3-generated subgroups tends top - 1:1, asn tends to infinity. This work is part of the author’s Ph.D. thesis carried out at the Hebrew University of Jerusalem under the supervision of Prof. A. Lubotzky. I wish to thank Prof. Lubotzky for his continual interest and encouragement without which this paper would not have been published.     

Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ? ²-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ? ^p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.     

Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques.In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.