Multilevel Evaluation of the General Dirichlet Series

Abstract

In this Study, an accurate method for summing the general Dirichlet series is presented. Long range terms of this series are calculated by a multilevel approach. The Dirichlet series, in this technique, is decomposed into two parts, a local part and a smooth part. The local part vanishes beyond some cut off distance, "$r_0$", and it can be cheaply computed . The complexity of calculations depends on $r_0$. The smooth part is calculated on a sequence of grids with increasing meshsize. Treating the smooth part using multilevels of grid points overcomes the high cost of calculating the long range terms. A high accuracy in approximating the smooth part is obtained with the same complexity of computing the local part. The method is tested on the Riemann Zeta function. Since there is no closed form for this function with odd integer orders, the method is applied for orders $s= 3, 5, 7,$ and $9$. In comparison with the direct calculations, remarkable results are obtained for $s=3$ and $s=5$; the reason is the major effect of the long range terms. For $s=7,$ and $s=9$, results obtained are better than those of direct calculations. The method is compared with efficient well known methods. The comparison shows the superiority of the multilevel method.