Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

The CO fundamental-band laser as secondary frequency standard at 5 µm

We present a very high-resolution heterodyne spectrometer based on a CO laser which operates down to fundamental-band transitions of the molecule. This allows us to detect saturated absorption signals on these transitions at very low pressure (0.4 Pa) and laser intensity (< 1 mW/cm²), yielding a linewidth of about 250 kHz. With the CO fundamental-band laser stabilized to these saturation signals we have measured the transition frequencies of the fundamental bands of three isotopic species to an accuracy of typically 20 kHz (v/v 3 × 10^–10), referenced to the CO₂ frequency standard. Together with additional frequency measurements of the first hot bands, these provide the first heterodyne frequency data of sub-Doppler accuracy for transitions in low lying bands of CO. They now represent the most accurate secondary frequency standard in the spectral region around 5 µm (60 THz).     

Formation of YAG from coprecipitated yttrium aluminium hydroxides

Thermal decomposition of the amorphous coprecipitate of yttrium and aluminium hydroxides forming yttrium aluminium garnet has been investigated employing thermal analyses, X-ray diffraction and IR spectroscopy. On heating, the coprecipitate progressively loses water forming a stable but highly disordered hydroxy garnet which crystallizes at 1180 K and decomposes to YAG at 1290 K. Nucleation of the crystalline phase appears to begin at 800 K.The authors thank Dr. P. V. Ravindran of the Analytical Chemistry and Shri N. K. Kulkarni of the Fuel Chemistry Divisions of BARC for their assistance in the thermoanalytical experiments.