Über die Primteiler-Anzahl ω(n)

It is shown by analytical means that, if one assumes the Riemann hypothesis, the asymptotic formula $$\sum\limits_{n \leqslant x} {\omega (n) = x 1n1n } x + B - x\int_l^{x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } {\frac{{\{ t\} }}{{t^2 (1n x - 1n t)}}dt + O(x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon } )} $$ holds. This improves a result ofB. Saffari, who got a weaker error term by using the Dirichlet “hyperbola method”. The above formula, in turn, implies the Riemann hypothesis.     

Γ-Lakton-cis-anellierung an Δ2- und Δ3- Cholesten

γ-Lactone-cis-annulation to Δ²- and Δ³- Cholestene. From Δ²- and Δ³- cholestene the γ-lactones 11a , 11b , 12a , and 12b are synthesized through the dibromocarbene adducts 3 and 4 , the bromohydrines 5 and 6 , the oxapiropentanes 7 and 8 , and the cyclobutanones 9a , 9b and 10a , 10b , respectively. The ¹³C-NMR.-spectra of 1–8 and 11 as well as the ORD.-spectra of the cyclobutanones 9 and 10 are reported.     

Transition metal nitrosyls as nitrosylation agents : II.Photo-nitrosylation of η5-C5H5V(CO)3L(L = CO,Pz3) by [Co(NO)2Br]2

η⁵-C₅H₅V(NO)₂CO is prepared in 40% yield by the photo-reaction between η⁵-C₅H₅V(CO)₄ and [Co(NO)₂Br]₂.η⁵-C₅H₅V(NO)₂CO reacts by an S_N1 mechanism with various phosphines PZ₃ to yield η⁵-C₅-H₅V(NO)₂PZ₃. The phosphine complexes are also obtained by photo-induced ligand interchange between η⁵-C₅H₅V(CO)₃PZ₃ and [Co(NO)₂Br]₂, or η⁵-C₅H₅V(CO)₄ and Co(NO)₂Br(PZ₃). In all cases, the main cobalt species formed is Co(NO)(CO)₃. While the one-bond vanadiumphosphorus coupling constants of most of the phosphine complexes are virtually the same (ca 410 Hz),the chemical shift values δ(⁵¹V) (?1328 to ?973 ppm rel. VOCl₃) decrease in the order PF₃ > CO > P(OR)₃ > P(alkyl)₃ > PPh₃ > PPh(NEt₂)₂, reflecting the decreasing π-acceptor ability of the ligands. δ(⁵¹V) also decreases in the series of alkylphosphines PR₃ (R = Me, Et, Prⁿ, Buⁱ, Prⁱ, BU^t) as the cone angle of PR₃increases.     

Perfluoralkyljod-verbindungen: Darstellungen und eigenschaften von CF3JO,CF3JOF2 und CF3JO2

Perfluoroalkyl iodine compounds: preparations and properties of CF₃IO, CF₃IOF₂, and CF₃IO₂. The trifluoromethyl iodine compounds CF₃IO, CF₃IOF₂, and CF₃IO₂ are formed from the reactions of CF₃I, CF₃IF₂ or CF₃IF₄ with ozone or silicon dioxide respectively. Their preparartions, properties, ¹⁹F-nmr spectra, and ir spectra are described.     

Untersuchungen über ternäre Chalkogenide. XXIV. Zur Struktur des TlGaSe2

On Ternary Chalcogenides. XXIV. The Structure of TlGaSe₂ TlGaSe₂ is monoclinic with a = 10.772(3), b = 10.771(5), c = 15.636(8) Å, ß = 100.6(3)°, Z = 16, space group Cc. The refinement yielded an R value of 0.082 for 1366 unique observed reflections (MoKα). The compound crystallizes with a layer structure with two anion layers perpendicular to c* in the unit cell. The structural motive of the layers are large corner-linked Ga₄Se₁₀- tetraeder consisting of four corner-linked GaSe₄ tetrahedra. Two adjacent layers are turned relative to each other by 90°. They are kept together by Tl¹⁺ ions, which are situated on straight lines parallel to the edges of the Ga₄Se₁₀ groups. They are surrounded by six selenium atoms forming trigonal-prismatic TlSe₆ polyhedra. 4₂ screw-axes within the layers are the reason for a strongly pronounced pseudo tetragonal symmetry. Different possibilities of stacking cause a onedimensional disorder along the c* direction. The relations between the TlGaSe₂ type and the TlSe type and also the high-pressure modifications of some compounds with the TlGaSe₂ structure are discussed.     

Enantioselektive Synthese von Allyl-, Propargyl- und 4-En-2-inyl-aminen durch 1,2-Addition von Organocer-Reagenzien an chirale Aldimine

Enantioselective Synthesis of Allyl-, Propargyl-, and 4-En-2-ynyl-amines via 1,2-Addition of Organocerium Reagents to Chiral Aldehyde Imines (E)- and (Z)-Allyl-, propargyl-, and 4-en-2-ynyl-amines 5 and 14 , useful bifunctional building blocks and of pharmaceutical interest, are synthesized in high enantiomeric purity (e.e. ≥ 97%). Key step is the diastereoselective 1,2-addition (d.e. 86 to ≥ 98%) of organocerium reagents to chiral α,β-unsaturated aldehyde imines 3 or 8 to produce adduct amines 4 and 9 (Schemes 1 and 4, resp.). The propargylamine 9 is a substrate for Pd-catalyzed coupling with alkenyl halides to produce the enynylamine 11a and the thienyl-substituted alkynylamine 11b . The chiral auxiliary (S,S)- 2 is removed from 4 and 11 in 3 steps affording the title compounds 5 and 14 . Diastereoisomer enrichment of the hydrochloride of 6 by crystallization is possible.     

The phenolic oxidation of a β-methylchalcone

The phenolic oxidation of 2',4-dihydroxy-4'-methoxy-β-methylchalcone using alkaline potassium ferricyanide gives an aurone rather than an isoflavone. This result is discussed in the context of current theories regarding the biosynthesis of flavonoid and isoflavonoid compounds.