On a trigonometric sum

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Zeta function regularization in Casimir effect calculations and J.S. Dowker's contribution

A summary of relevant contributions, ordered in time, to the subject of operator zeta functions and their application to physical issues is provided. The description ends with the seminal contributions of Stephen Hawking and Stuart Dowker and collaborators, considered by many authors as the actual starting point of the introduction of zeta function regularization methods in theoretical physics, in particular, for quantum vacuum fluctuation and Casimir effect calculations. After recalling a number of the strengths of this powerful and elegant method, some of its limitations are discussed. Finally, recent results of the so called operator regularization procedure are presented.Comment: 16 pages, dedicated to J.S. Dowker, version to appear in International Journal of Modern Physics

Zeta-Function Regularization is Uniquely Defined and Well

Hawking's zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading lore about different difficulties associated with it. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method ---and other analytical procedures--- are clarified and corrected.Comment: 7 pages, LaTeX fil

Explicit Zeta Functions for Bosonic and Fermionic Fields on a Noncommutative Toroidal Spacetime

Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function, in particular, the residua of the poles and their finite parts are explicitly given there. An important novelty is the fact that simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.Comment: 15 pages, no figures, LaTeX fil

Vacuum energy for the supersymmetric twisted D-brane in constant electromagnetic field

We calculate vacuum energy for twisted SUSY D-brane on toroidal background with constant magnetic or constant electric field. Its behaviour for toroidal D-brane (p=2) in constant electric field shows the presence of stable minimum for twisted versions of the theory. That indicates such a background maybe reasonable groundstate.Comment: LaTeX, 10 page

Principal forms X^2 + nY^2 representing many integers

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hambur

Zeta function regularization for a scalar field in a compact domain

We express the zeta function associated to the Laplacian operator on $S^1_r\times M$ in terms of the zeta function associated to the Laplacian on $M$, where $M$ is a compact connected Riemannian manifold. This gives formulas for the partition function of the associated physical model at low and high temperature for any compact domain $M$. Furthermore, we provide an exact formula for the zeta function at any value of $r$ when $M$ is a $D$-dimensional box or a $D$-dimensional torus; this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated to the physical models at finite temperature.Comment: 19 pages, no figures, to appear in J. Phys.

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

Primary Hepatic Lymphoma: A Retrospective, Multicenter Rare Cancer Network Study.

Primary hepatic lymphoma (PHL) is a rare malignancy. We aimed to assess the clinical profile, outcome and prognostic factors in PHL through the Rare Cancer Network (RCN). A retrospective analysis of 41 patients was performed. Median age was 62 years (range, 23-86 years) with a male-to-female ratio of 1.9:1.0. Abdominal pain or discomfort was the most common presenting symptom. Regarding B-symptoms, 19.5% of patients had fever, 17.1% weight loss, and 9.8% night sweats. The most common radiological presentation was multiple lesions. Liver function tests were elevated in 56.1% of patients. The most common histopathological diagnosis was diffuse large B-cell lymphoma (65.9%). Most of the patients received Chop-like (cyclophosphamide, doxorubicin, vincristine, and prednisone) regimens; 4 patients received radiotherapy (dose range, 30.6-40.0 Gy). Median survival was 163 months, and 5- and 10-year overall survival rates were 77 and 59%, respectively. The 5- and 10-year disease-free and lymphoma-specific survival rates were 69, 56, 87 and 70%, respectively. Multivariate analysis revealed that fever, weight loss, and normal hemoglobin level were the independent factors influencing the outcome. In this retrospective multicenter RCN study, patients with PHL had a relatively better prognosis than that reported elsewhere. Multicenter prospective studies are still warranted to establish treatment guidelines, outcome, and prognostic factors

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space Hn, new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions