2010/08/30 by Bi, Guangqing, Bi, Yuekai
#11M06 #11M35 #35S99 #42A24 #Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1008.5046
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).