scholarly journals Dedekind Sums for a Fuchsian Group, I

1973 ◽  
Vol 50 ◽  
pp. 21-47 ◽  
Author(s):  
Larry Joel Goldstein

The well-known first limit formula of Kronecker asserts thatwhere z = x + iy is contained in the complex upper halfplane H, C = the Euler-Mascheroni constant, and η(z) is the Dedekind eta-function defined by

1974 ◽  
Vol 53 ◽  
pp. 235-237 ◽  
Author(s):  
Larry Joel Goldstein
Keyword(s):  

1974 ◽  
Vol 53 ◽  
pp. 171-187 ◽  
Author(s):  
Larry Joel Goldstein

In [1] we derived a generalization of Kronecker’s first limit formula. Our generalization was a limit formula for the Eisenstein series for an arbitrary cusp of a Fuchsian group Γ of the first kind operating on the complex upper half-plane H. In that work, we introduced Dedekind sums associated to the principal congruence subgroups Γ(N) of the elliptic modular group. The work of our preceding paper suggests a natural question: Is there a generalization of Kronecker’s second limit formula to the setting of a general Fuchsian group of the first kind? The answer to this question is the subject of this paper.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.


2015 ◽  
Vol 12 (02) ◽  
pp. 1550019
Author(s):  
Emilio Elizalde ◽  
Klaus Kirsten ◽  
Nicolas Robles ◽  
Floyd Williams

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla–Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yuankui Ma ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Hanyoung Kim ◽  
Taekyun Kim

AbstractThe classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind-type DC sums, which are obtained from the Dedekind-type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.


Author(s):  
L. Alayne Parson

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)


Author(s):  
K.K. SEKHRI ◽  
C.S. ALEXANDER ◽  
H.T. NAGASAWA

C57BL male mice (Jackson Lab., Bar Harbor, Maine) weighing about 18 gms were randomly divided into three groups: group I was fed sweetened liquid alcohol diet (modified Schenkl) in which 36% of the calories were derived from alcohol; group II was maintained on a similar diet but alcohol was isocalorically substituted by sucrose; group III was fed regular mouse chow ad lib for five months. Liver and heart tissues were fixed in 2.5% cacodylate buffered glutaraldehyde, post-fixed in 2% osmium tetroxide and embedded in Epon-araldite.


Author(s):  
Anton Bózner ◽  
Mikuláš Gažo ◽  
Jozef Dostál

It is anticipated that Japanese quail /Coturnix coturnix japonica/ will provide animal proteins in long term space flights. Consequently this species of birds is of research interest of international space program INTERCOSMOS. In the year 1987 we reported on an experiment /2/ in which the effect of chronic acceleration of 2 G hypergravitation, the hypodynamy and the simultaneous effect of chronic acceleration and the location in the centre of the turntable of the centrifuge on the protein fractions in skeletal muscles was studied. The ultrastructure of the heart muscle was now in this experiments examined as well.Japanese quail cockerels, aged 48 days were exposed to 2 G hypergravitation /group IV/ in a 6,4 m diameter centrifuge, to hypodynamy /group III/ and their combination /group V/, respectively for 6 days / Fig.1/. The hypodynamy in group III was achieved by suspending the birds in jackets without contact the floor. The group II was located in the centre ofthe turntable of the centrifuge. The control group I. was kept under normal conditions. The quantitative ultrastructure of myocard was evaluated by the methods of Weibel/3/ - this enables to determine the number, relative size and volume of mitochondria volume of single mitochondria, defficiency of mitochondrial cristae and volume of myofibrils.


1998 ◽  
Vol 80 (09) ◽  
pp. 393-398 ◽  
Author(s):  
V. Regnault ◽  
E. Hachulla ◽  
L. Darnige ◽  
B. Roussel ◽  
J. C. Bensa ◽  
...  

SummaryMost anticardiolipin antibodies (ACA) associated with antiphospholipid syndrome (APS) are directed against epitopes expressed on β2-glycoprotein I (β2GPI). Despite a good correlation between standard ACA assays and those using purified human β2GPI as the sole antigen, some sera from APS patients only react in the latter. This is indicative of heterogeneity in anti-β2GPI antibodies. To characterize their reactivity profiles, human and bovine β2GPI were immobilized on γ-irradiated plates (β2GPI-ELISA), plain polystyrene precoated with increasing cardiolipin concentrations (CL/β2GPI-ELISA), and affinity columns. Fluid-phase inhibition experiments were also carried out with both proteins. Of 56 selected sera, restricted recognition of bovine or human β2GPI occurred respectively in 10/29 IgA-positive and 9/22 IgM-positive samples, and most of the latter (8/9) were missed by the standard ACA assay, as expected from a previous study. Based on species specificity and ACA results, IgG-positive samples (53/56) were categorized into three groups: antibodies reactive to bovine β2GPI only (group I) or to bovine and human β2GPI, group II being ACA-negative, and group III being ACA-positive. The most important group, group III (n = 33) was characterized by (i) binding when β2GPI was immobilized on γ-irradiated polystyrene or cardiolipin at sufficient concentration (regardless of β2GPI density, as assessed using 125I-β2GPI); (ii) and low avidity binding to fluid-phase β2GPI (Kd in the range 10–5 M). In contrast, all six group II samples showed (i) ability to bind human and bovine β2GPI immobilized on non-irradiated plates; (ii) concentration-dependent blockade of binding by cardiolipin, suggesting epitope location in the vicinity of the phospholipid binding site on native β2GPI; (iii) and relative avidities approximately 100-fold higher than in group III. Group I patients were heterogeneous with respect to CL/β2GPI-ELISA and ACA results (6/14 scored negative), possibly reflecting antibody differences in terms of avidity and epitope specificity. Affinity fractionation of 23 sera showed the existence, in individual patients, of various combinations of antibody subsets solely reactive to human or bovine β2GPI, together with cross-species reactive subsets present in all samples with dual reactivity namely groups III and II, although the latter antibodies were poorly purified on either column. Therefore, the mode of presentation of β2GPI greatly influences its recognition by anti-β2GPI antibodies with marked inter-individual heterogeneity, in relation to ACA quantitation and, possibly, disease presentation and pathogenesis.


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