The zeros of the Riemann zeta-function

Abstract

The infra-red spectra of a considerable number of carboxylic acids and their COOD derivatives have been investigated between 1500 and 500 cm$^{-1}$, as dimeric units in the liquid or crystalline solid states. Under these conditions the COOH group is shown usually to give rise to strong absorption bands in the regions 1420 $\pm $ 20, 1300 $\pm $ 15 and 935 $\pm $ 15 cm$^{-1}$. The first two of these are found to correspond to closely coupled OH deformation and C$\chembond{1,0} $O stretching vibrations occurring in the plane of the (COOH)$_{2}$ dimeric ring; the latter is caused by the out-of-plane OH deformation vibration. COOD groups have absorption bands in the ranges 1350 $\pm $ 50, 1050 $\pm $ 10 and 675 $\pm $ 25 cm$^{-1}$, which can be assigned respectively to the C$\chembond{1,0} $O stretching mode and the in-plane and out-of-plane OD deformation vibrations. Less constant absorption bands of the COOH group between 700 and 575 cm$^{-1}$ are attributed to O$\chembond{1,0} $C$\chembond{2,0} $O skeletal deformation vibrations, and the corresponding bands are found at slightly lower frequencies in the spectra of the COOD derivatives. The results of the infra-red investigation are compared with the Raman spectra of such acids and with related infra-red and Raman frequencies of other molecules. Infra-red spectra of some equimolecular mixtures of acids with water have also been studied.