New Properties of Fourier Series and Riemann Zeta Function

Abstract

We establish the mapping relations between analytic functions and periodic functions using the abstract operators cos(h∂x) and sin(h∂x), including the mapping relations between power series and trigonometric series, and by using such mapping relations we obtain a general method to find the sum function of a trigonometric series. According to this method, if each coefficient of a power series is respectively equal to that of a trigonometric series, then if we know the sum function of the power series, we can obtain that of the trigonometric series, and the non-analytical points of which are also determined at the same time, thus we obtain a general method to find the sum of the Dirichlet series of integer variables, and derive several new properties of ζ(2n + 1).

A New Operator Theory of Linear Partial Differential Equations

“…Proof. (See [6]) For sin(h∂ x ) arctan bx and cos(h∂ x ) arctan bx, using (19), ( 21) and (25) we can obtain…”

Recursion Relations and Functional Equations for the Riemann Zeta Function

“…Recursion relations with summations are handled e.g. in [2], [3], [4], [5], [6], [7] and [8] and thus outside the scope of this work. Some new ideas have appeared too.…”