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1: 25.6 Integer Arguments
§25.6(i) Function Values
25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
§25.6(ii) Derivative Values
§25.6(iii) Recursion Formulas
For related results see Basu and Apostol (2000).
2: 25.12 Polylogarithms
The remainder of the equations in this subsection apply to principal branches. … The special case z = 1 is the Riemann zeta function: ζ ( s ) = Li s ( 1 ) . … Further properties include …and … In terms of polylogarithms …
3: 6.16 Mathematical Applications
Hence, if x is fixed and n , then S n ( x ) 1 4 π , 0 , or 1 4 π according as 0 < x < π , x = 0 , or π < x < 0 ; compare (6.2.14). … It occurs with Fourier-series expansions of all piecewise continuous functions. … … If we assume Riemann’s hypothesis that all nonreal zeros of ζ ( s ) have real part of 1 2 25.10(i)), then …where π ( x ) is the number of primes less than or equal to x . …
4: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
§25.11(i) Definition
The Riemann zeta function is a special case: …
§25.11(ii) Graphics
§25.11(vi) Derivatives
5: Software Index
‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.

  • 6: 26.12 Plane Partitions
    A plane partition is self-complementary if it is equal to its complement. …
    §26.12(ii) Generating Functions
    §26.12(iii) Recurrence Relation
    where ζ is the Riemann ζ -function25.2(i)). Addendum: To ten decimal places, …
    7: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • J. van de Lune, H. J. J. te Riele, and D. T. Winter (1986) On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.
  • I. M. Vinogradov (1958) A new estimate of the function ζ ( 1 + i t ) . Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • H. von Koch (1901) Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).
  • 8: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • M. V. Berry (1995) The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • J. M. Borwein, D. M. Bradley, and R. E. Crandall (2000) Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121 (1-2), pp. 247–296.
  • P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.
  • 9: Bibliography D
  • N. G. de Bruijn (1937) Integralen voor de ζ -functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).
  • C. de la Vallée Poussin (1896a) Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction ζ ( s ) de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • T. M. Dunster (2006) Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
  • 10: Bibliography
  • V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.
  • G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.