Mean values of certain zeta-functions on the critical line

Department of Mathematics, University of Belgrade (Yugoslavia). Department of Mathematics, University of Naples (Italy). Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 4, pp. 701–714, October–December, 1989.     

On the Error Term for the Fourth Moment of the Riemann Zeta-Function

Let E₂(T) denote the error term in the asymptotic formula for^T₀|(+it)|⁴dt. It is proved that there exist constants A>0,B>1 such that for TT₀>0 every interval [T, BT] containspoints T₁, T₂ for which and that ^T₀|E₂(t)|^adt>>T^1+(a/2) for any fixed a1. Theseresults complement earlier results of Motohashi and Ivi that^T₀E₂(t)dt<<T^3/2 and that ^T₀E²₂(t)dt<<T²log^CT forsome C>0. Omega-results analogous to the above ones are obtainedalso for the error term in the asymptotic formula for the Laplacetransform of |(+it)|⁴.     

The vibron dressing in α-helicoidal macromolecular chains

We present a study of the physical properties of the vibrational excitation in α-helicoidal macromolecular chains,caused by the interaction with acoustical and optical phonon modes. The influence of the temperature and the basic systemparameters on the vibron dressing have been analyzed by employing the simple mean-field approach based on the variationalextension of the Lang-Firsov unitary transformation. The applied approach predicts a region in system parameter spacewhere one has an abrupt transition from a partially dressed (light and mobile) to a fully dressed (immobile) vibron state.We found that the boundary of this region depends on system temperature and the type of bond among structural elementsin the macromolecular chain.