An analogue to a conjecture of S. Chowla and H. Walum

It is proved that (for every ε > 0)

$\text{∑}\text{n?T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{∑n<}\text{T}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{n}{}^{\mathrm{a}}\text{m}{}^{\mathrm{b}}\text{B}{}_{\mathrm{k}}\text{({}\text{T}\text{nm}\text{}) = O(T}{}^{\mathrm{<mtext>(a+b+1)</mtext><mtext>3?</mtext>}}\text{)}$

(where {·} denotes the fractional part and B_k the Bernoulli polynomial of order k) under the suppositions that k ≥ 2 and 2a ? 1 ≥ b ≥ 1. If (1) were true for k = 1, a = b = 0, then Piltz' divisor problem (for n = 3) would be readily solved. This is an analog to a conjecture formulated by S. Chowla and H. Walum in 1963 and settled in the affirmative (under suitable suppositions) quite recently by S. Kanemitsu and R. Sita Rama Chandra Rao.     

A general method for the preparation of N-monosubstituted-3-isothiazolamines

We describe here a novel and general high yield method for the synthesis of 3-alkyl and arylaminoisothiazoles, a previously unknown group of isothiazoles, by the reaction of ammonia with 3-halo-2-alkylisothiazolium salts. The mechanism of the reaction probably involves a ring opening followed by a recyclization to the title compounds.     

The Henicosaphosphide Iodides of Potassium and Rubidium,K4P21I and Rb4P21I

The novel ternary polyphosphides M₄P₂₁I (M = K, Rb) have been synthesized from the elements in single crystalline form, representing further examples for the formation of mixed crystals between simple salts and binary phosphides. They form as ruby‐red platelets and dark‐red prisms, respectively, and are only slightly sensitive to moisture and oxygen. The compounds are isotypic (Ccmm (no 63); Z = 4; oP104; K₄P₂₁I: a = 12.853Å; b = 21.795Å; c = 9.748Å; 1168 hkl, R = 0.033; Rb₄P₂₁I: a = 13.281Å; b = 21.868Å; c = 9.771Å; 777 hkl, R = 0.053) and feature corrugated 2D networks formed from two different types of polymerized P₇ units. The networks form large cavities filled by M⁺ and I^‐ ions. Zigzag chains of condensed trigonal M₆ prisms, centered by the I^‐ anions, separate the polyphosphide nets. The mean homoatomic P‐P bond length (d = 2.216Å) corresponds to a P‐P single bond. However, the individual P‐P distances vary with position and function (2.126 ‐ 2.247Å) and these are compared with those of the isolated P₂₁^‐3 anion.     

Lattice points in a circle and divisors in arithmetic progressions

Let A(x) denote the number of lattice points in the circle u²+v²x and define as the infimum of all reals for which . The objective of this paper is to show that 35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.     

Die Struktur des Pentaphenyldisilans

Structure of Pentaphenyldisilane For the first time Pentaphenyldisilane was prepared by Gilman and Goodman. It is produced by the reaction of Ph₃SiLi with Ph₂ClSiH. The crystal structure presents an ideally staggered conformation. The distance d(Si? Si) = 235.7 pm corresponds to a normal single bond length. This emphasizes the complete relief of the central Si? Si bond by the insertion of only one hydrogen atom.     

Coordination around thorium(IV) in aqueous perchlorate,chloride and nitrate solutions

The coordination around the thorium(IV) ion in aqueous perchlorate, chloride and nitrate solutions has been determined from large angle X-ray scattering measurements. In perchlorate solutions, where inner-sphere complexes are not formed, the first coordination sphere contains 8.0±0.5 water molecules with Th-H₂O bond lengths of 2.48₅ Å. In chloride solutions inner-sphere complexes are formed, which lead to an increase in the coordination number. In nitrate solutions the nitrate ions are bonded as bidentate ligands to the thorium ion. The bond lengths are similar to those found in crystalline hydrates of thorium nitrate. The coordination numbers found for thorium(IV) in solution are compared with previously reported values for lower charged ions of similar size.On leave from Department of Inorganic Chemistry Royal Institute of Technology S-10044 Stockholm Sweden     

Multiple bonds between main group elements and transition metals, 155. (Hexamethylphosphoramide) methyl(oxo) bis(η-peroxo)rhenium(VII), the first example of an anhydrous rhenium peroxo complex: crystal structure and catalytic properties

Methyl(oxo)bis(η²-peroxo)rhenium(VII)1, the active species of the system CH₃ReO₃/H₂O₂ in the catalytic oxidation of different organic and organometallic compounds, is stabilized by a water molecule attached to the rhenium center. This water molecule can be removed and substituted by hexamethylphosphoramide (HMPA) to yield (hexamethylphosphoramide)methyl(oxo)bis(η²-peroxo rhenium(VII) (3). The synthesis, crystal structure (X-ray difraction study), and catalytic properties of which compound are reported. Crystal data are as follows: monoclinic, space group P2₁/n, A = 900.76(7) pm, B = 1229.80(11) pm, C = 1318.57(11) pm, β = 90.251(7)°, R_w = 0.034 for 1878 reflections. The catalytic properties of compound 3 in the oxidation of olefins with H₂O₂ are similar to those of 1.     

Structure determination of zinc iodide complexes formed in aqueous solution

Structures of the complexes formed in aqueous solutions between zinc(II) and iodide ions have been determined from large-angle X-ray scattering, Raman and far-IR measurements. The coordination in the hydrated Zn²⁺ hexaaqua ion and the first iodide complex, [ZnI]⁺, is octahedral, but is changed into tetrahedral in the higher complexes, [ZnI₂(H₂O)₂], [ZnI₃(H₂O)]^– and [ZnI₄]^2–. The Zn-I bond length is 2.635(4)Å in the [ZnI₄]^2– ion and slightly shorter, 2.592(6)Å, in the two lower tetrahedral complexes. In the octahedral [ZnI(H₂O)₅]⁺ complex the Zn-I bond length is 2.90(1)Å. The Zn-O bonding distances in the complexes are approximately the same as that in the hydrated Zn²⁺ ion, 2.10(1)Å.     

Intensitätsverhalten von Molekülanionen organischer Substanzen unter verschiedenen Versuchsbedingungen

A modified commercial mass spectrometer was used to perform quantitative measurements on negative ions of selected organic compounds at about 10^?4 Torr source pressure. The pressure dependency of the molecular ion intensity on pure compounds and binary mixtures shows up two different sources of slow secondary electrons. At low total source pressures a log-log plot of the ion intensity against the sample amount is linear and slow electrons are produced predominantly by wall effects, whereas at high pressures plasma effects arise with a non-linear pressure dependency.