Extreme values of the Dedekind Ψ\Psi function

Let $\Psi(n):=n\prod_{p | n}(1+\frac{1}{p})$ denote the Dedekind $\Psi$ function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{\Psi(n)}{n\log\log n}.$ We prove unconditionally that $R(n)< e^\gamma$ for $n\ge 31.$ Let $N_n=2...p_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\frac{e^\gamma}{\zeta(2)}$ for $n\ge 3$ is equivalent to the Riemann Hypothesis.Comment: 5 pages, to appear in Journal of Combinatorics and Number theor

Areas of Seal R/D at GE

About four years ago, work was completed on a 36 inch diameter gas to gas carbon ring seal used to buffer low pressure turbine air at the rim of the forward outer flowpath on the GE36 unducted fan (UDF) engine. At about the same time, we were developing a long life counter-rotating intershaft air-oil seal of approximately 7.6 inch diameter for operation at 800 fps, 800 F, and 50 psid. Although we were successful in meeting most program goals with a split ring seal of the axial bushing type, the seal with the greatest payoff in life and air leakage rates, bearing many features in common with the GE36 seal, could not be successfully tested because of the structural weakness of the primary seal ring carbon material. This was a split ring seal using a hybrid combination of orifice compensated hydrostatic and shrouded hydrodynamic gas bearings. We are presently working to develop this design in conjunction with high strength materials being developed by Pure Carbon Co. In the area of engine secondary gas flow path-sealing for performance improvement, we are currently working with carbon and all metal face seals. A 15 inch diameter all metal 'aspirating' face seal, using self-acting hydrostatic bearings, was successfully tested to 700 fps, 100 psid, and 1000 F, demonstrating long life at flow reduction of 86 percent compared to a 'best' labyrinth. This seal will be developed through 1400 F, 900 fps, and 350 psid. The seal 'aspirates' closed at about idle speed pressure during engine start and reopens at engine shutdown. A hydraulic thrust balance seal, currently using orifice compensated hydrostatics, is under development. Other aspects of these projects are briefly covered

Containerless processing of Nb-Ge alloys in a long drop tube

The thirty-two meter drop tube at the Marshall Space Flight Center was used to study the effect of zero gravity containerless processing on the structure and properties of materials. The concept involves the suppression of heterogeneous nucleation of solid in liquid and, therefore, solidification accompanied by large degrees of undercooling. Under these conditions metastable phases can be formed or, at the very least, unique nonequilibrium microstructures (containing equilibrium phases) with unique properties can be produced. The drop tube solidification was applied to niobium base alloys with emphasis on the Nb-Ge binary system in an effort to produce metastable phases with high superconducting transition temperatures in bulk specimens. In the past, only lower Ge alloys (Nb-13 a/o, Nb-18 a/o, and Nb-22 a/o) could be undercooled. Higher Ge alloys (e.g., Nb-25 a/o Ge and Nb-27 a/o Ge) can now be undercooled on a routine basis