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Multilevel Evaluation of the General Dirichlet Series

Year 2020, Volume: 4 Issue: 4, 443 - 458, 30.12.2020
https://doi.org/10.31197/atnaa.810766

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.

References

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  • B. Wang, A. Iserles, Dirichlet series for dynamical systems of first-order ordinary differential equations, Discrete and Continuous Dynamical Systems Series B, Volume 19, Number 1, (2014).
  • A. Pickering, J. Prada, Dirichlet series and the integrability of multilinear differential equations, Journal of Mathematical Physics 46, 043504, (2005).
  • E.C. Titchmarsh, The Theory of the Riemann Zeta Function. Claredon Press, Oxford, (1986).
  • R. Beals and R. Wong,Specialfunctions,Cambridge University Press, 40-43, (2010).
  • J. M. Borwein, D. M. Bradley and R. E. Cradall, Computational strategies for the Riemann zeta function. J. Comput. Appl. Math, 121, 247-296, (2000).
  • P. Borwein, An efficient algorithm for the Riemann Zeta function, Canadian Mathematical Society Conference Proceedings, 27:29-34, (2000.)
  • H. M. Srivastava, Some simple algorithms for evaluations and representations of the Riemann zeta function at positive integer arguments, J. Math. Anal. Appl, 246, 331-351,(2000) .
  • J. M. Borwein, D. M. Bradley and R. E. Crandall, “Com-putational Strategies for the Riemann Zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8, (2000).
  • Yu. V. Matiyasevich, Riemann's zeta function and finite Dirichlet series, St. Petersburg Math. J. 27, 985-1002, (2016).
  • G. Sudhaamsh, M. Reddy and S. S. Rau, Some Dirichlet Series and Means of Their Coefficients, Southeast Asian Bulletin of Mathematics, 40: 585–591, (2016)..
  • Yu. V. Matiyasevich, A Few Factors from the Euler Product Are Sufficient for Calculating the Zeta Function with High Precision, Proceedings of the Steklov Institute of Mathematics, 299: 178–188, (2017).
  • A. Defant, I. Schoolmann, Variants of a theorem of Helson on general Dirichlet series, Journal of Functional Analysis, Volume 279, Issue 5, (2020).
  • Yu. V. Matiyasevich, In: Fillion N., Corless R., Kotsireas I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. (2019).
  • Yu. V. Matiyasevich, Plausible ways for calculating the Riemann zeta function via the Riemann–Siegel theta function, Journal of Number Theory, Volume 207, 460-471, (2020).
  • J. R. Poulin, Calculating Infinite Series Using Parseval's Identity, Phd thesis, (2020).
  • D. Benko, The Basel problem as a telescoping series. College Math. J. 43, No. 3, 244-250, (2012).
  • Zh.-X. Wang and D.-R. Guo, T`esh¯u H´ansh`u G`ail`un (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, (Chinese), (2000).
  • D. Cvijovic, J. Klinowski, Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math, 142, 435-439 (2002).
  • K. Dilcher, Asymptotic behavior of Bernoulli, Euler, and generalized Bernoulli polynomials, Journal of approximation theory, 49, 321-330, (1987).
  • J. L. Lopez, N. M. Temme, Large degree asymptotics of generalized Bernoulli and Euler polynomials, J. Math. Anal. Appl, 363, 197-208, (2010).
  • J. Choi, Rapidly converging series for $\zeta(2n + 1)$ from Fourier seires, Abstract and Applied Analysis, Vol. 2014, Available at: http: //dx.doi.org/10.1155/ 2014/ 457620, (2014).
  • F. M. S. Lima, A simpler proof of a Katsuradas theorem and rapidly converging series for $\zeta(2n + 1)$ and $\zeta(2n)$. arXiv: 1203.5660v2.
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  • Juuso T. Olkkonen, Hannu Olkkonen, Accelerated Series for Riemann Zeta Function at Odd Integer Arguments, Open Journal of Discrete Mathematics, 3, 18-20, (2013).
  • R. Ap´ery: Irrationalit´e de $\zeta(2)$ et $\zeta(3)$, in Journ´ees arithm´etiques (Luminy, 1978), Ast´erisque 61, 11–13, (1979).
  • F. Beukers: A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11, no. 3, 268–272, (1979).
  • K. Ball, T. Rivoal: Irrationalit´e d’une infinit´e de valeurs de la fonction zˆeta aux entiers impairs, Invent. Math. 146, no. 1, 193–207, (2001).
  • D. Huylebrouck: Similarities in irrationality proofs for p, $ln 2$, $\zeta(2)$, and $\zeta(3)$, Amer. Math. Monthly 108 , no. 3, 222–231, (2001).
  • A. van der Poorten: A proof that Euler missed . . . Ap´ery’s proof of the irrationality of $\zeta(3)$, Math. Intelligence, 1 , no. 4, 195–203,(1978/79) .
  • T. Rivoal: La fonction zˆeta de Riemann prend une infinit´e de valeurs irrationnelles aux entiers impairs, C.R. Acad. Sci. Paris S´er. I Math. 331 , no. 4, 267–270, (2000).
  • V.V. Zudilin: One of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, Russian Math. Surveys 56 , no. 4,774–776 ,(2001) .
  • Madelung, E., Das Elektrsche feld in systemen von regelmabig angeordneten punktladungen. Phys. Zeits, 19, 524, (1918).
  • Ewald, P. P., Die berechnung optischer und elektrostatischer gitterpotentiale. Ann. Phys., (1921), 363, 253-287. Yakub, E., Ronchi, C., A new method for computation of long ranged coulomb forces in computer simulation of disordered systems. J. Low Temp.,139, 633, (2005).
  • Yakub, E., Ronchi, C., An efficient method for computation of long ranged coulomb forces in computer simulation of ionic fluids, J. Chem. Phys.,, 119, 11556,(2003) .
  • Hunenburger, P.H., McCammon, J. A., Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: A continuum electrostatic study, Biophys. Chem., 78, 69, (1999).
  • Tsukerman, I., Efficient computation of long range electromagnetic interactions without Fourier Transforms, IEEE Trans. Magn., 40, 2158, (2004).
  • Ghasemi, S. A., Neelov, A., Geodecker, S., A particle-particle, particle density algorithm for the calculation of electrostatic interactions of particles with slablike geometry, J. Chem. Phys., 127, 24102-24105, (2007).
  • Neelov, A., Ghasemi, S. A., Geodecker, S., particle-particle, particle scaling function algorithm for the electrostatic problems in free boundary conditions, J. Chem. Phys.,, 127, 24109-24116, (2007).
  • Poplau, G., Rienen, U. V., Geer, B.V.D., Loos, M.D, Multigrid algorithms for the fast calculation of space-charge effects in accelerator design. IEEE Trans. Magn., 40, 714-717,(2004) .
  • Brandt, A., Multilevel computations of integral transforms and particle interactions wit oscillatory kernels. Comput. Phys. Commun., 56, 24-38, (1991).
  • Suwan, I., Multiscale methods in molecular dynamics, PhD thesis, weizmann Institute of Science, Israel, Rehovot, (2006).
  • Hardy, D. J., Stone J. E., Schulten, K., Multilevel summation of electrostatic potentials using graphics processing units, Parallel Comput., (2009), 35, 164-177. Richard P. Brent and Paul Zimmermann, Modern Computer Arithmetic, (2010).
Year 2020, Volume: 4 Issue: 4, 443 - 458, 30.12.2020
https://doi.org/10.31197/atnaa.810766

Abstract

References

  • A. Pickering, Dirichlet Series for Quasilinear Partial Differential Equations, Theoretical and Mathematical Physics, 135(2):638-641, (2003).
  • A. Pickering, On the Identification of Integrable Equations Using Dirichlet Series, Progress of Theoretical Physics, Volume 108, Issue 3, 603-607, (2002).
  • B. Wang, A. Iserles, Dirichlet series for dynamical systems of first-order ordinary differential equations, Discrete and Continuous Dynamical Systems Series B, Volume 19, Number 1, (2014).
  • A. Pickering, J. Prada, Dirichlet series and the integrability of multilinear differential equations, Journal of Mathematical Physics 46, 043504, (2005).
  • E.C. Titchmarsh, The Theory of the Riemann Zeta Function. Claredon Press, Oxford, (1986).
  • R. Beals and R. Wong,Specialfunctions,Cambridge University Press, 40-43, (2010).
  • J. M. Borwein, D. M. Bradley and R. E. Cradall, Computational strategies for the Riemann zeta function. J. Comput. Appl. Math, 121, 247-296, (2000).
  • P. Borwein, An efficient algorithm for the Riemann Zeta function, Canadian Mathematical Society Conference Proceedings, 27:29-34, (2000.)
  • H. M. Srivastava, Some simple algorithms for evaluations and representations of the Riemann zeta function at positive integer arguments, J. Math. Anal. Appl, 246, 331-351,(2000) .
  • J. M. Borwein, D. M. Bradley and R. E. Crandall, “Com-putational Strategies for the Riemann Zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8, (2000).
  • Yu. V. Matiyasevich, Riemann's zeta function and finite Dirichlet series, St. Petersburg Math. J. 27, 985-1002, (2016).
  • G. Sudhaamsh, M. Reddy and S. S. Rau, Some Dirichlet Series and Means of Their Coefficients, Southeast Asian Bulletin of Mathematics, 40: 585–591, (2016)..
  • Yu. V. Matiyasevich, A Few Factors from the Euler Product Are Sufficient for Calculating the Zeta Function with High Precision, Proceedings of the Steklov Institute of Mathematics, 299: 178–188, (2017).
  • A. Defant, I. Schoolmann, Variants of a theorem of Helson on general Dirichlet series, Journal of Functional Analysis, Volume 279, Issue 5, (2020).
  • Yu. V. Matiyasevich, In: Fillion N., Corless R., Kotsireas I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. (2019).
  • Yu. V. Matiyasevich, Plausible ways for calculating the Riemann zeta function via the Riemann–Siegel theta function, Journal of Number Theory, Volume 207, 460-471, (2020).
  • J. R. Poulin, Calculating Infinite Series Using Parseval's Identity, Phd thesis, (2020).
  • D. Benko, The Basel problem as a telescoping series. College Math. J. 43, No. 3, 244-250, (2012).
  • Zh.-X. Wang and D.-R. Guo, T`esh¯u H´ansh`u G`ail`un (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, (Chinese), (2000).
  • D. Cvijovic, J. Klinowski, Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math, 142, 435-439 (2002).
  • K. Dilcher, Asymptotic behavior of Bernoulli, Euler, and generalized Bernoulli polynomials, Journal of approximation theory, 49, 321-330, (1987).
  • J. L. Lopez, N. M. Temme, Large degree asymptotics of generalized Bernoulli and Euler polynomials, J. Math. Anal. Appl, 363, 197-208, (2010).
  • J. Choi, Rapidly converging series for $\zeta(2n + 1)$ from Fourier seires, Abstract and Applied Analysis, Vol. 2014, Available at: http: //dx.doi.org/10.1155/ 2014/ 457620, (2014).
  • F. M. S. Lima, A simpler proof of a Katsuradas theorem and rapidly converging series for $\zeta(2n + 1)$ and $\zeta(2n)$. arXiv: 1203.5660v2.
  • C. Nash, and D. J. O'Connor, J. Math. Phys. 36, pp. 1462-1505, (1995).
  • Juuso T. Olkkonen, Hannu Olkkonen, Accelerated Series for Riemann Zeta Function at Odd Integer Arguments, Open Journal of Discrete Mathematics, 3, 18-20, (2013).
  • R. Ap´ery: Irrationalit´e de $\zeta(2)$ et $\zeta(3)$, in Journ´ees arithm´etiques (Luminy, 1978), Ast´erisque 61, 11–13, (1979).
  • F. Beukers: A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11, no. 3, 268–272, (1979).
  • K. Ball, T. Rivoal: Irrationalit´e d’une infinit´e de valeurs de la fonction zˆeta aux entiers impairs, Invent. Math. 146, no. 1, 193–207, (2001).
  • D. Huylebrouck: Similarities in irrationality proofs for p, $ln 2$, $\zeta(2)$, and $\zeta(3)$, Amer. Math. Monthly 108 , no. 3, 222–231, (2001).
  • A. van der Poorten: A proof that Euler missed . . . Ap´ery’s proof of the irrationality of $\zeta(3)$, Math. Intelligence, 1 , no. 4, 195–203,(1978/79) .
  • T. Rivoal: La fonction zˆeta de Riemann prend une infinit´e de valeurs irrationnelles aux entiers impairs, C.R. Acad. Sci. Paris S´er. I Math. 331 , no. 4, 267–270, (2000).
  • V.V. Zudilin: One of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, Russian Math. Surveys 56 , no. 4,774–776 ,(2001) .
  • Madelung, E., Das Elektrsche feld in systemen von regelmabig angeordneten punktladungen. Phys. Zeits, 19, 524, (1918).
  • Ewald, P. P., Die berechnung optischer und elektrostatischer gitterpotentiale. Ann. Phys., (1921), 363, 253-287. Yakub, E., Ronchi, C., A new method for computation of long ranged coulomb forces in computer simulation of disordered systems. J. Low Temp.,139, 633, (2005).
  • Yakub, E., Ronchi, C., An efficient method for computation of long ranged coulomb forces in computer simulation of ionic fluids, J. Chem. Phys.,, 119, 11556,(2003) .
  • Hunenburger, P.H., McCammon, J. A., Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: A continuum electrostatic study, Biophys. Chem., 78, 69, (1999).
  • Tsukerman, I., Efficient computation of long range electromagnetic interactions without Fourier Transforms, IEEE Trans. Magn., 40, 2158, (2004).
  • Ghasemi, S. A., Neelov, A., Geodecker, S., A particle-particle, particle density algorithm for the calculation of electrostatic interactions of particles with slablike geometry, J. Chem. Phys., 127, 24102-24105, (2007).
  • Neelov, A., Ghasemi, S. A., Geodecker, S., particle-particle, particle scaling function algorithm for the electrostatic problems in free boundary conditions, J. Chem. Phys.,, 127, 24109-24116, (2007).
  • Poplau, G., Rienen, U. V., Geer, B.V.D., Loos, M.D, Multigrid algorithms for the fast calculation of space-charge effects in accelerator design. IEEE Trans. Magn., 40, 714-717,(2004) .
  • Brandt, A., Multilevel computations of integral transforms and particle interactions wit oscillatory kernels. Comput. Phys. Commun., 56, 24-38, (1991).
  • Suwan, I., Multiscale methods in molecular dynamics, PhD thesis, weizmann Institute of Science, Israel, Rehovot, (2006).
  • Hardy, D. J., Stone J. E., Schulten, K., Multilevel summation of electrostatic potentials using graphics processing units, Parallel Comput., (2009), 35, 164-177. Richard P. Brent and Paul Zimmermann, Modern Computer Arithmetic, (2010).
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Iyad Suwan 0000-0002-8657-1327

Publication Date December 30, 2020
Published in Issue Year 2020 Volume: 4 Issue: 4

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