Multilevel Evaluation of the General Dirichlet Series

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 o distance, "r0", and it can be cheaply computed . The complexity of calculations depends on r0. 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 e ect 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 e cient well known methods. The comparison shows the superiority of the multilevel method.


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where t i , s are complex numbers and λ(i) is a strictly increasing sequence of positive numbers that tends to infinity. The function (1) is considered to be one of the most important special functions in physics, mathematics, and engineering. In particular, Dirichlet series plays important role in dierential and integral equations, for example [1]- [4]. Hence, developing fast and accurate techniques for approximating this function has been the focus of numerous researches during the last and the present centuries as in [7]- [17]. A very important and famous derivation of (1) is the Riemann Zeta function which is denes as follows: If λ(i) = ln(i) and t i = (−1) i−1 , then (1), becomes The Riemann Zeta function is dened by [8] ζ(s) = 1 For even integers, Euler [18] has found a closed form of (3), where B n is the n th Bernoulli number [19]. Obviously, the more precise the approximated value of π is, the more accurate value of ζ(2k) can be obtained. For odd positive integers, no closed form is known, therefore, many analytical and numerical methods have been derived in order to accurately evaluate Riemann Zeta functions. An integral expression is known for these functions [20]: where B 2k+1 (x) are Bernoulli polynomials [21,22]. The closed form of the integral in (5) is still an open problem. The Fourier analysis and Katsuradas theorem are used to construct a rapidly converging series for ζ(2k + 1) in [23]and [24] respectively. A remarkable expression for ζ(2k + 1) has been derived in [25], and accelerated series for those functions is found in [26,27]. Considerable irrationality results for values of Riemann Zeta functions at odd positive integers are obtained and presented in [27]- [33]. In fact, hundreds of researches have been carried out in both analytical and numerical approaches in order to shed light on the convergence of the Riemann Zeta functions and their applications. The advantage of any of these methods over one another is determined by the amount of work needed to reach a desired accuracy, and the speed of convergence. The accuracy of direct calculations of (1) depends on the number of terms involved. Considering the long range terms certainly gives accurate results of the approximation, but this will increase the complexity of computations. Fast and accurate calculation of the long range part of innite summations, in general, had been the subject of many researches from the beginning of the last century. The rst eective summation methods were carried out by Madelung and Ewald in [34] and [35] respectively. Those methods were applied to sum the long range electrostatic interactions of a crystalline lattice. Since then, many improvements have been done on Ewald's method as in [36]- [38]. Alternative methods were introduced during the past decades in order to reduce the computational work in calculating the long range part of innite summations, for example [39]- [42]. In this paper, a multilevel technique has been developed and presented in order to avoid the slow direct summation. The general multilevel approach in the context of general transformation, many body problems, and matrix multiplication has been initially proposed by Brandt in [43]. The approach is also used to develop the multilevel Monte-Carlo simulation method in statistical mechanics [44]. The method facilitates developing computational tools that describe, scale by scale, a material property at increasingly larger scales [45]. In the frame of the method presented in this paper, the Riemann Zeta function is decomposed into two components, a local part and a smooth part. The local part vanishes beyond some cut-o radius r cut and is computed directly and cheaply. The smooth part is accurately calculated on a sequence of grids with very low computer demand. In the next section of this paper, the developed scheme is presented in details, and the method is tested on Riemann Zeta functions at n = 3, 5, 7 and 9. In section 3, the multilevel technique results are compared with those of two well known methods in [8] and [46]. Conclusions and future perspectives are presented in section 4.

Analysis of the Method
The analysis of the method in this paper starts from the following equivalent denition of the Dirichlet series (1): where t i = t −i and In the frame of the multilevel method, the kernel (7) is split into two parts: where and G sm (i) = P m (i) , |i| ≤ r 0 e −sλ(|i|) , |i| > r 0 The cut o distance r 0 is a positive real number, and P m is a polynomial of the form The part (9) is comprised of short-range inuences, and it is called "local". It can be computed in a constant number of operations for a given r 0 . The part (10) is called "smooth" and it satises the assumption The coecients c j in (11) are determined by applying the smoothness criterion (12). For simplicity, the values of the coecients c j in (11) can be universalized by changing the variable |i| into and where I. Suwan, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 443458. 446 and the coecients a j are determined by the assumption Changing the variables from |i| to |i| r 0 is of great importance in calculations; it reduces the computational work because the values of a j , in this case, can be found and saved in a pre-calculated table regardless the value of r 0 . Now, using formulas (13) and (14), (6) can be expressed in the form where and The notation [...] refers to the greatest integer function. It is easy to see that calculating U loc in (18) is computationally cheap for small values of r 0 .The main purpose of the method presented in this paper is accurately approximating the smooth part U sm with computational work that is linear function of that of the local part. The rst step of the method is rewriting equation (19) by adding and subtracting the value This value denotes the "self contribution" of the computations, where x 0 = 0. This contribution is calculated once during the procedure. Using (20), equation (19) becomes where and Omitting E(r 0 ) from (22) gives the rst approximation of (1), where I. Suwan, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 443458.
More accurate approximation of f (s) can be achieved by rening the value ofŨ self sm (s) as an approximation of U self sm (s). This can be done by constructing a successive nite number L of coarse levels of equidistance integer points. At each level l = 1, ....., L, the distance between any two neighboring grid points is I l = 2 l−1 , Table (1) shows the distances between grid points for l = 1, 2, and 3. The construction of the coarse levels can be done by successive splitting of the kernel G in (7) into local and smooth parts at each level l ≥ 1 as follows: At each coarse Level l, if we let the cut-o distance R(l) = 2 l−1 r 0 , then the number of grid points inside R(l) is the same in all levels, and the distance between any two neighboring grid points at level l is 2 l−1 . Hence, from (27) where Equation (28) can be written in the formŨ where C = I. Suwan, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 443458.

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The constant C is the same for all levels. So, letting L approaches infinity and summing the innite geometric series in (30) givesŨ Using (18), (20), and (32), the nal approximation of f (s) is ).
Once the constant C in (31) is found, the complexity of approximating f (s) in (33) is the same as in (18) with additional constant term. Finding C itself is done one time for a given r 0 , and its computational work is also the same as (18). Therefore, the accuracy and the whole complexity in approximating f (s) in (33) is linearly proportional to the complexity of calculating the local part in (18). The number of grid points involved in calculation are the same in all levels because the cut-o distance is doubled in each coarse level in comparison with the ner one, and the distance between any two neighboring points are also doubled. The gain is that by increasing the level, more long range terms are involved in calculations.

Testing the Method
In this section, we will test the eciency of the presented method on the Reimann Zeta function. Denition (3) can be written in the form, where the kernel G is For a given cut-o distance r 0 = r cut , from (33) and (34), the approximated Zeta function is and the constant C is given by where The method is tested on approximating the Riemann Zeta functions at s = n, where n = 3, 5, 7, and 9.   Direct calculations of (34) requires ignoring the terms beyond some cut-o distance, say, the r cut used in calculating (36). In this case, the comparison between the approximation and the direct calculations makes sense. For this purpose, the following "direct" zeta function is dened: Table (9) shows ζ(n) using the direct calculations (39) up to 12 digits and shows the r cut needed to reach this accuracy. For the purpose of examining the accuracy of the multilevel method, the relative error is calculated considering the values of Zeta functions in table (9) as "exact" values. For n = 3, Figures (1), (2), (3) show the direct evaluation, and the approximation using m = 2, 4, and 6 respectively. Figure (4) shows the percentage relative error in calculating ζ(3) using m = 2. Figure (5) shows the percentage relative error in calculating ζ(3) using m = 2, 4, and 6. It is easy to notice that the approximated solution is more accurate than the direct calculations, and the accuracy increases by increasing m for small values of r cut . Figure (6) shows the approximation and the direct calculations for ζ(5) using m = 6. Figures (7), and (8) show the error in calculating ζ(7) and ζ(9) using m = 2 and m = 4 respectively. In all cases, for small values of r cut , the error is very small in comparison with the direct calculations.
Table (10) shows the r cut needed in multilevel technique in order to reach the same results in table (9). One can easily see the eciency of the multilevel method.      Figure 1: ζ(3) using Multilevel approach with m=2, the direct calculations, and the "exact" value Table 9: The exact value of ζ(n) up to 12 digits and the rcut used in (39) n ζ(n) r cut 3 1.202056903159 more than 100000 5 1.036927755143 1173 7 1.008349277381 108 9 1.002008392826 32

Comparison with ecient Algorithms
In this section, numerical results of the multilevel method is compared with those of two well known methods that are used to eciently calculate Zeta functions at positive integer variables. The two methods are listed below. M ethod1 [46]: In this method, the Zeta function may be evaluated to any desired precision if m and r cut = p are chosen large enough in the Euler-Maclaurin formula:  where and B 2k are the Bernoulli numbers. Table( 11) shows ζ(n) for m = 100 at integer values of r cut between 1 and 10. Figure (9) shows the convergence of the multilevel method and method 1 for ζ(3). Comparing results in table (11) with those of the multilevel approach in tables (6), (7), and (8) shows the superiority of the multilevel method. M ethod2 [8]: For P m (x) = m k=0 a k x k , an arbitrary polynomial of degree m that does not vanish at −1. If then The suggested polynomial in [8] is p(x) = x n (1−x) n . Table ( 12) shows the values of Zeta function at dierent values of the polynomial degree m in (43). It is easy to see that the multilevel results for m = 2 is better than those of method 2 using at least m = 10. Of course increasing m in (43) will increase the accuracy but this increases the computational work, while the multilevel approach is of very low computational work for small values of m and r cut .

Conclusions and Discussion
Accurate evaluation of general Dirichlet series is important in many problems in science and engineering. For this purpose, a multilevel approach is developed in this paper and is successfully applied. In the frame of this approach, the function (6) is split into two parts. The rst of which is a local part (9) comprising of short range inuences, and it can therefore be computed in a constant number of operations. The second part, (10), satises a well-dened smoothness criterion and is computed in a cheap and ecient way. The calculation of (10) is carried out recursively for increasingly coarser grids, the distance between each two neighboring grid points at each coarse level is doubled in comparison to the ner one. Theoretically, The recursion proceeds till innity which leads to a convergent geometric series (30). The sum of it is constant. In fact, the Dirichlet series is split into a sum of several parts each of which is local at some level. A self contribution (20) is added and subtracted from (19) which completes the calculations of the smooth part. The convergence of the method is tested on a kind of Dirichlet series, the Riemann Zeta function ζ(n) for n = 3, 5, 7, and 9. The method converges independently of n. In comparison with direct calculations, the convergence becomes faster and the accuracy becomes better by decreasing n. For small values of n, the eect of the long range terms becomes larger and the method works more eciently. This fact comes from the methodology of the method designed in order to calculate innite summations with long range properties. Hence, this procedure can be applied in physical problems involving simulations of long range potentials which will be an alternative method of the well known Ewald's technique. The technique is compared with well known ecient methods. The comparison shows that the multilevel technique is distinctive. As an application of the method, Monte Carlo simulations of systems containing long range potentials can be performed in faster and ecient way. This will be the base of our future plan.