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1: 25.10 Zeros
Also, ζ ( s ) 0 for s = 1 , a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). … Riemann developed a method for counting the total number N ( T ) of zeros of ζ ( s ) in that portion of the critical strip with 0 < t < T . By comparing N ( T ) with the number of sign changes of Z ( t ) we can decide whether ζ ( s ) has any zeros off the line in this region. … Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of ζ ( s ) in the critical strip are on the critical line (van de Lune et al. (1986)). More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …
2: Bibliography B
  • H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.
  • R. J. Baxter (1982) Exactly Solved Models in Statistical Mechanics. Academic Press Inc., London-New York.
  • M. V. Berry (1991) Infinitely many Stokes smoothings in the gamma function. Proc. Roy. Soc. London Ser. A 434, pp. 465–472.
  • W. G. C. Boyd (1987) Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
  • H. M. Bui, B. Conrey, and M. P. Young (2011) More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.
  • 3: Errata
  • Subsection 25.10(ii)

    In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

  • Notation

    In §3.7(iii), the symbol 𝐀 P is now being used in several places instead of 𝐀 in order to disambiguate symbols.

  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Poor spacing in math was corrected in several chapters.

  • Equation (16.15.3)

    In applying changes in Version 1.0.12 to (16.15.3), an editing error was made; it has been corrected.

  • 4: Brian D. Sleeman
     1939 in London, U. …  in 1966 at the University of London where he was advised by Felix Arscott (1922-1996). … He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. … Plank) of Differential equations and mathematical biology, published by CRC Press in 2003, with a second edition in 2010. Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …
    5: Robb J. Muirhead
     1946 in Adelaide, South Australia) is Senior Director, Statistical Research and Consulting Center, Pfizer Global R&D, New London, Connecticut. …in Mathematics and Ph. …in Statistics were awarded by the University of Adelaide in 1967 and 1970, respectively. … His book Aspects of Multivariate Statistical Theory was published by John Wiley & Sons in 1982. Muirhead was elected a Fellow of the American Statistical Association in 1994. …
    6: Michael V. Berry
     1941 in Frimley, U. … Berry has received many awards for his work in physics. He was elected a Fellow of the Royal Society of London in 1982, and of the Royal Society of Edinburgh in 2005. He was knighted in 1996. …National Academy of Sciences in 1995, and of the Royal Netherlands Academy of Arts and Sciences in 2000. …
    7: Peter L. Walker
     1942 in Dorchester, U. …He began his academic career in 1964 at the University of Lancaster, U. … Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … Walker was elected a member of the London Mathematical Society (1970). Walker is now retired and living in Cheltenham, UK. …
    8: Christopher J. Howls
     1966 in Greenwich, U. …is Senior Lecturer in Applied Mathematics at the University of Southampton, U. … Howls has published numerous papers in the areas of asymptotics and its applications. …Institute of Mathematics and Its Applications in 2007. …In 2008 he was appointed to the editorial board of the Proceedings of the Royal Society of London Series A. …
    9: Frank W. J. Olver
     1924 in Croydon, U. …degrees in mathematics from the University of London in 1945, 1948, and 1961, respectively. … In 1992 he retired, and was appointed Professor Emeritus. … , University College London, U. … In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. …
    10: Bibliography Q
  • F. Qi (2008) A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.
  • S.-L. Qiu and J.-M. Shen (1997) On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.