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 $\cos (h{\partial }_x)$ 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 $\zeta (2n+1)$.

arXiv comment28 pages

References in corpus (3)

• Some infinite series involving the Riemann zeta function
• A New Operator Theory of Linear Partial Differential Equations
• Stationary Solutions of the Klein-Gordon Equation in a Potential Field

Cited by in corpus (5)

• A New Operator Theory of Linear Partial Differential Equations
• Solutions of Cauchy problem for multiple inhomogeneous wave equation
• Recursion Relations and Functional Equations for the Riemann Zeta Function
• On Evaluation of Zeta and Related Functions by Abstract Operators
• Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study