New Properties of Fourier Series and Riemann Zeta Function

TL;DR

This paper introduces a novel method using abstract operators to relate analytic and periodic functions, enabling new insights into Fourier series and deriving properties of the Riemann zeta function at odd integers.

Contribution

It develops a general approach to find sums of trigonometric series from power series and uncovers new properties of (2n+1) using this method.

Findings

Established mapping relations between analytic and periodic functions.

Developed a method to find sums of trigonometric series from power series.

Derived new properties of (2n+1).

Abstract

We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, 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 $\zeta(2n+1)$.