We review the present status of the dynamical study of the many-electron problem and introduce the dynamical theory of response and relaxation through the continued fraction formalism using the frequency moments developed by the projection technique or the recurrence relation method. We apply this theory to the electron gas system with homogeneous positive background. We calculate the higher order frequency moments at a large momentum transfer for the first time and obtain an asymptotically exact solution at a particular metallic density in terms of these frequency moments. Our dynamic structure factor describes the experiment quantitatively well. Through the dynamical study, we obtain a momentum independent relation which may test the degree of precision of a theory.
II. Density Response and Relaxation
III. Continued Fraction Representation
IV. Frequency Moments
V. Exact Solution at Large-k
VI. Concluding Remarks