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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). … The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
2: 27.15 Chinese Remainder Theorem
This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. For example, suppose a lengthy calculation involves many 10-digit integers. Most of the calculation can be done with five-digit integers as follows. … Even though the lengthy calculation is repeated four times, once for each modulus, most of it only uses five-digit integers and is accomplished quickly without overwhelming the machine’s memory. …
3: 25.18 Methods of Computation
Calculations relating to derivatives of ζ ( s ) and/or ζ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). … Most numerical calculations of the Riemann zeta function are concerned with locating zeros of ζ ( 1 2 + i t ) in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of ζ ( s ) lie on the critical line s = 1 2 . Calculations to date (2008) have found no nontrivial zeros off the critical line. …
4: 25.17 Physical Applications
The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …
5: 26.22 Software
  • Inverse Symbolic Calculator (website).

  • 6: 31.18 Methods of Computation
    Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of z ; see Laĭ (1994) and Lay et al. (1998). …
    7: Bibliography V
  • A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.
  • A. L. Van Buren, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..
  • A. L. Van Buren and J. E. Boisvert (2002) Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.
  • A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.
  • Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
  • 8: Bibliography Y
  • A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov (1985) The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.
  • 9: Bibliography K
  • M. K. Kerimov and S. L. Skorokhodov (1984a) Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.
  • M. Kodama (2011) Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
  • G. C. Kokkorakis and J. A. Roumeliotis (1998) Electromagnetic eigenfrequencies in a spheroidal cavity (calculation by spheroidal eigenvectors). J. Electromagn. Waves Appl. 12 (12), pp. 1601–1624.
  • K. S. Kölbig, J. A. Mignaco, and E. Remiddi (1970) On Nielsen’s generalized polylogarithms and their numerical calculation. Nordisk Tidskr. Informationsbehandling (BIT) 10, pp. 38–73.
  • E. Konishi (1996) Calculation of complex polygamma functions. Sci. Rep. Hirosaki Univ. 43 (1), pp. 161–183.
  • 10: 27.18 Methods of Computation: Primes
     Oliveira e Silva has calculated π ( x ) for x = 10 23 , using the combinatorial methods of Lagarias et al. (1985) and Deléglise and Rivat (1996); see Oliveira e Silva (2006). …