Complete asymptotic expansions associated with Epstein zeta-functions

Let Q(u,v)=|u+vz|² be a positive-definite quadratic form with a complex parameter z=x+iy in the upper-half plane. The Epstein zeta-function attached to Q is initially defined by for Re s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of as y→+∞ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition 1 in Sect. 3). This procedure, differs slightly from other previously known methods of the analytic continuation, gives a new alternative proof of the Fourier expansion of (Proposition 2 in Sect. 3). The use of Mellin-Barnes type of integral formulae is crucial in all aspects of the proofs; several transformation properties of hypergeometric functions are especially applied with manipulation of these integrals. Research supported in part by Grant-in-Aid for Scientific Research (No. 13640041), the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

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.     

关于广义Dedekind和的加权均值

利用Dirichlet L－函数的均值定理和特征和估计，研究了广义Dedekind和与HurwitzZeta-函数的加权均值分布性质，并给出一个有趣的渐近公式。     

On well-rounded sublattices of the hexagonal lattice

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.     

On the definity of quadratic forms subject to linear constraints

A short proof is given of the necessary and sufficient conditions for the positivity and nonnegativity of a quadratic form subject to linear constraints.     

Joint value-distribution theorems on Lerch zeta-functions. II

We give corrected statements of some theorems from [5] and [6] on joint value-distribution of Lerch zeta-functions (limit theorems, universality, functional independence). We also present a new direct proof of a joint limit theorem in the space of analytic functions and an extension of a joint universality theorem. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 332–350, July–September, 2006.     

Dedekind zeta-functions and Dedekind sums

In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n³ -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n² + 1, and the class number of quadratic number field K2 = Q(vq) is 1.     

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

On the secondary zeta-functions

Using summability it is shown that $\text{∑}{}_{\mathrm{n}}\text{?2 (Λ(n) ? 1) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(log n)}{}^{\mathrm{?8}}$ defines an entire function in the s-plane. Its asymptotic nature is found and a functional equation relating it to the series $\text{∑{i(}\text{1}\text{2}\text{? p)}}{}^{\mathrm{1?8}}$, Im p = γ > 0,is obtained where p = β + iγ are the nontrivial zeros of Riemann's zeta-function.     

On the stability and nonexistence of turing patterns for the generalized Lengyel‐Epstein model

This paper studies the dynamics of the generalized Lengyel‐Epstein reaction‐diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady‐state solution.     

On bilinear forms in normal variables

Bilinear forms in normal variables when the matrices of the forms are rectangular are considered. Explicit expressions for the cumulants, joint cumulants and joint cumulants of bilinear and quadratic forms are given. Necessary and sufficient conditions are established for the independence of two bilinear forms as well as a bilinear and a quadratic form. Special cases are shown to agree with known results.     

On the 3-Pfister Number of Quadratic Forms

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F^×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d²)/8 3-fold Pfister forms.     

On some slowly convergent series involving the Hurwitz zeta-function

We shall extract the essence of the Adamchik–Srivastava generating function method (Analysis (Munich) 18 (1998) 131) by proving the most far-reaching Ramanujan–Yoshimoto formula and by showing that some of the results stated in Srivastava and Choi (Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001) are simple consequences of the above-mentioned formula.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

On the mean square of Lerch zeta-functions

Using the approximate functional equation for L(l,a, s) = ?_n=0^￥ [(e(ln))/((n+a)^s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.     

Yuan's alternative theorem and the maximization of the minimum eigenvalue function

LetA ₁ andA ₂ be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionxR ⁿ max {x ^T A ₁ x,x ^T A ₂ x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices.This research was partially supported by Dirección General de Investigación Científica y Técnica (DGICYT) under Project PB92-0615.     

On splitting of totally singular quadratic forms

In this note we completely study the standard splitting of quasi-Pfister forms and their neighbors, and we include some general results on standard splitting towers of totally singular quadratic forms. The author was supported by the European research network HPRN-CT-2002-00287 “AlgebraicK-Theory, Linear Algebraic Groups and Related Structures”.