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1: 8.22 Mathematical Applications
If ζ x ( s ) denotes the incomplete Riemann zeta function defined by
8.22.2 ζ x ( s ) = 1 Γ ( s ) 0 x t s - 1 e t - 1 d t , s > 1 ,
so that lim x ζ x ( s ) = ζ ( s ) , then
8.22.3 ζ x ( s ) = k = 1 k - s P ( s , k x ) , s > 1 .
For further information on ζ x ( s ) , including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …
2: Bibliography K
  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
  • K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.
  • 3: Bibliography D
  • T. M. Dunster (2006) Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
  • 4: Bibliography V
  • 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.
  • H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • 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. von Koch (1901) Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).
  • 5: Bibliography
  • M. M. Agrest and M. S. Maksimov (1971) Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.
  • 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.
  • H. Alzer (1997b) On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.
  • 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.
  • 6: Bibliography L
  • L. Levey and L. B. Felsen (1969) On incomplete Airy functions and their application to diffraction problems. Radio Sci. 4 (10), pp. 959–969.
  • N. Levinson (1974) More than one third of zeros of Riemann’s zeta-function are on σ = 1 2 . Advances in Math. 13 (4), pp. 383–436.
  • J. S. Lew (1994) On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.
  • X. Li, X. Shi, and J. Zhang (1991) Generalized Riemann ζ -function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.
  • L. Lorch and P. Szegő (1964) Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.
  • 7: 5.11 Asymptotic Expansions
    and … where ζ ( K ) is as in Chapter 25. In the case K = 1 the factor 1 + ζ ( K ) is replaced with 4. … For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). …
    8: Errata
  • Source citations

    Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

  • Equation (19.25.37)

    The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

  • Equation (25.2.4)

    The original constraint, s > 0 , was removed because, as stated after (25.2.1), ζ ( s ) is meromorphic with a simple pole at s = 1 , and therefore ζ ( s ) - ( s - 1 ) - 1 is an entire function.

    Suggested by John Harper.

  • Chapter 25 Zeta and Related Functions

    A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

  • Equation (26.12.26)
    26.12.26 pp ( n ) ( ζ ( 3 ) ) 7 / 36 2 11 / 36 ( 3 π ) 1 / 2 n 25 / 36 exp ( 3 ( ζ ( 3 ) ) 1 / 3 ( 1 2 n ) 2 / 3 + ζ ( - 1 ) )

    Originally this equation was given incorrectly as

    pp ( n ) ( ζ ( 3 ) 2 11 n 25 ) 1 / 36 exp ( 3 ( ζ ( 3 ) n 2 4 ) 1 / 3 + ζ ( - 1 ) ) .

    Reported 2011-09-05 by Suresh Govindarajan.

  • 9: 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.
  • B. C. Berndt (1972) On the Hurwitz zeta-function. Rocky Mountain J. Math. 2 (1), pp. 151–157.
  • 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.
  • 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.
  • 10: Bibliography R
  • H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.
  • R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).
  • B. Riemann (1859) Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monats. Berlin Akad. November 1859, pp. 671–680.
  • B. Riemann (1899) Elliptische Functionen. Teubner, Leipzig.
  • B. Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen.