New Properties of Fourier Series and Riemann Zeta Function

arXiv1008.5046MaRDI QIDQ6220439

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Publication date: 30 August 2010

Abstract: We establish the mapping relations between analytic functions and periodic functions using the abstract operators $cos(hpartia{l}_x)$ and $sin(hpartia{l}_x)$, 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 $zeta(2n+1)$.

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