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1: 25.10 Zeros
More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …
2: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • M. Brack, M. Mehta, and K. Tanaka (2001) Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.
  • 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).

  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

  • References

    Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

  • 4: 24.20 Tables
    Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. In Wagstaff (2002) these results are extended to n = 60 ( 2 ) 152 and n = 40 ( 2 ) 88 , respectively, with further complete and partial factorizations listed up to n = 300 and n = 200 , respectively. …
    5: 25.3 Graphics
    See accompanying text
    Figure 25.3.1: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , 20 x 10 . Magnify
    See accompanying text
    Figure 25.3.3: Modulus of the Riemann zeta function | ζ ( x + i y ) | , 4 x 4 , 10 y 40 . Magnify 3D Help
    6: 27.2 Functions
    Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …
    Table 27.2.1: Primes.
    n p n p n + 10 p n + 20 p n + 30 p n + 40 p n + 50 p n + 60 p n + 70 p n + 80 p n + 90
    Table 27.2.2: Functions related to division.
    n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
    1 1 1 1 14 6 4 24 27 18 4 40 40 16 8 90
    2 1 2 3 15 8 4 24 28 12 6 56 41 40 2 42
    7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
    7: 26.2 Basic Definitions
    Table 26.2.1: Partitions p ( n ) .
    n p ( n ) n p ( n ) n p ( n )
    3 3 20 627 37 21637
    6 11 23 1255 40 37338
    8: 28.35 Tables
  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ( 1 ) 6 , q = 0 ( 1 ) 10 ( 2 ) 20 ( 4 ) 40 ; 7D. Also ce n ( x , q ) , se n ( x , q ) for q = 0 ( 1 ) 10 , x = 1 ( 1 ) 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = 𝑏𝑒 n 2 q , b n = 𝑏𝑜 n 2 q .

  • Kirkpatrick (1960) contains tables of the modified functions Ce n ( x , q ) , Se n + 1 ( x , q ) for n = 0 ( 1 ) 5 , q = 1 ( 1 ) 20 , x = 0.1 ( .1 ) 1 ; 4D or 5D.

  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ( q ) , b n + 1 ( q ) for n = 0 ( 1 ) 4 , q = 0 ( 1 ) 50 ; n = 0 ( 1 ) 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ( 50 ) 200 . Fourier coefficients for ce n ( x , 10 ) , se n + 1 ( x , 10 ) , n = 0 ( 1 ) 7 . Mathieu functions ce n ( x , 10 ) , se n + 1 ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , x = 0 ( 5 ) 90 . Modified Mathieu functions Mc n ( j ) ( x , 10 ) , Ms n + 1 ( j ) ( x , 10 ) , and their first x -derivatives for n = 0 ( 1 ) 4 , j = 1 , 2 , x = 0 ( .2 ) 4 . Precision is mostly 9S.

  • Ince (1932) includes the first zero for ce n , se n for n = 2 ( 1 ) 5 or 6 , q = 0 ( 1 ) 10 ( 2 ) 40 ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small q .

  • 9: Bibliography R
  • J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • D. St. P. Richards (Ed.) (1992) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.
  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • D. H. Rouvray (1995) Combinatorics in Chemistry. In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.), pp. 1955–1981.
  • 10: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • C. G. Kokologiannaki, P. D. Siafarikas, and C. B. Kouris (1992) On the complex zeros of H μ ( z ) , J μ ( z ) , J μ ′′ ( z ) for real or complex order. J. Comput. Appl. Math. 40 (3), pp. 337–344.
  • I. M. Krichever and S. P. Novikov (1989) Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).