Integral representations of the Riemann zeta function for odd-integer arguments

AbstractWe deduce four new integral representations for ζ(2n+1),n∈N, where ζ(s) is the Riemann zeta function

catena-[1,3-diammoniopropane di-mu2-hydroxodi-mu4-phosphato-trioxotrivanadium dihydrate]: a redetermination at 180 (2) K

The crystal structure of the title compound, (C3H12N2)- [V3O3(OH)2(PO4)2] 2H2O, has been reported by Soghomonian et al. [Chem. Mater. (1993), 5, 1690±1691]. We present here a redetermination of greatly improved precision and at a low temperature of 180 (2) K. The H atoms connected to oxygen have been successfully located and the coordination environments of the two crystallographically independent vanadium centres have been properly elucidated. Large channels, running along the a direction, contain water molecules and 1,3-diammoniopropane cations that are strongly hydrogen bonded to the anionic framework through N+ÐH O and OÐH O interactions. One vanadyl (V O) bond and the central ±CH2± group of 1,3-diammoniopropane are located on a mirror plane

Tetrabutylammonium 2,6-dihydroxybenzoate 2,6-dihydroxybenzoic acid solvate

The title compound, C16H36N+ C7H5O4 ÿ C7H6O4 or (nBu4N)+ (2,6-dhb) ÿ (2,6±Hdhb) [where (nBu4N) is tetrabutylammonium, (2,6-dhb) ÿ is 2,6-dihydroxybenzoate and (2,6-Hdhb) is 2,6-dihydroxybenzoic acid], crystallizes in the space group P1. The crystal structure involves alternate layers, in the c direction, of cationic [nBu4N]n n+ and anionic [(2,6- dhb)(2,6-Hdhb)]n nÿ . Hydrogen bonds within the anionic layers form bridges between 2,6-Hdhb and 2,6-dhb ÿ residues

Poly[[μ2-aqua-tetraaquahexakis(μ4-naph­thalene-2,6-dicarboxylato)tetra­holmium(III)] 1.75-hydrate]

In the title compound, {[Ho4(C12H6O4)6(H2O)5]·1.75H2O}n, which is isostructural with its Tb3+- and Eu3+-containing analogues, there are four crystallographically independent Ho3+ centres, each exhibiting a highly distorted HoO8 bicapped trigonal-prismatic coordination environment. Adjacent polyhedra are inter­connected via the carboxyl­ate groups and one μ2-bridging water mol­ecule, forming one-dimensional chains propagating along [100]. The naphthalene-2,6-dicarboxylate ligands further inter­connect these chains into a three-dimensional framework, which has zigzag channels housing the water mol­ecules. Two naphthalene-2,6-dicarboxylate bridging ligands have their centroids located on crystallographic centres of inversion. One water O atom has a fixed site occupancy factor of 0.75

1-Hydr­oxy-1,1,3,3,3-penta­phenyl­disiloxane, [Si2O(OH)(Ph)5], at 150 K

In the crystal structure of the title compound, C30H26O2Si2, one Si(Ph)3 residue is bound to another Si(OH)(Ph)2 residue via a nonlinear Si—O—Si bridge. The asymmetric unit is composed of four [Si2O(OH)(Ph)5] molecules. Each pair of adjacent molecules inter­acts via strong and highly directional O—H⋯O hydrogen bonds connecting neighbouring Si—OH units, and via inter-unit O—H⋯π contacts connecting the second hydroxyl groups with adjacent phenyl groups

Bis(tetra­methyl­amonium) bis­(2,4,5-carboxy­benzoate)–benzene-1,2,4,5-tetra­carboxylic acid (1/1)

The asymmetric unit of the title compound, 2C4H12N+·2C10H5O8 −·C10H6O8, consists of a tetra­methyl­amonium cation, an anion derived from the singly deprotonated pyromellitic acid anion, 2,4,5-carboxy­benzoate (H3bta−), and one-half of a benzene-1,2,4,5-tetra­carboxylic acid (H4bta) mol­ecule, which has the centroid of the aromatic ring positioned at a crystallographic centre of inversion. The H4bta and H3bta− residues are involved in an extensive inter­molecular O—H⋯O hydrogen-bonding network, which leads to a three-dimensional supra­molecular structure containing one-dimensional channels running parallel to the [001] crystallographic direction. These channels house the tetra­methyl­amonium cations

Decaaqua­dioxidobis[μ3-N-(phospho­n­atometh­yl)imino­diacetato]­dizinc(II)­divanadium(IV) dihydrate

The title compound, [Zn2V2(C5H6NO7P)2O2(H2O)10]·2H2O, contains a [V2O2(pmida)2]4− dimeric anionic unit [where H4pmida is N-(phosphono­meth­yl)imino­diacetic acid] lying on a centre of symmetry which is exo-coordinated via the two deprotonated phospho­nate groups to two Zn2+ cations, with the coordination environment of Zn completed by five water mol­ecules. The crystal packing is mediated by an extensive network of strong and highly directional O—H⋯O hydrogen bonds involving the water mol­ecules (coordinated and uncoordinated) and the functional groups of pmida4−, leading to a three-dimensional supra­molecular network