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1: 25.18 Methods of Computation
§25.18(ii) Zeros
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. …
2: 25.10 Zeros
§25.10(i) Distribution
In the region 0 < s < 1 , called the critical strip, ζ ( s ) has infinitely many zeros, distributed symmetrically about the real axis and about the critical line s = 1 2 . … Calculations relating to the zeros on the critical line make use of the real-valued function … 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)). …
3: 25.17 Physical Applications
Analogies exist between the distribution of the zeros of ζ ( s ) on the critical line and of semiclassical quantum eigenvalues. …
4: 8.22 Mathematical Applications
The function Γ ( a , z ) , with | ph a | 1 2 π and ph z = 1 2 π , has an intimate connection with the Riemann zeta function ζ ( s ) 25.2(i)) on the critical line s = 1 2 . …
5: 25.15 Dirichlet L -functions
There are also infinitely many zeros in the critical strip 0 s 1 , located symmetrically about the critical line s = 1 2 , but not necessarily symmetrically about the real axis. …
6: Bibliography B
  • 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.
  • 7: 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).

  • 8: 31.15 Stieltjes Polynomials
    The zeros z k , k = 1 , 2 , , n , of the Stieltjes polynomial S ( z ) are the critical points of the function G , that is, points at which G / ζ k = 0 , k = 1 , 2 , , n , where …
    9: Bibliography
  • M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.
  • B. H. Armstrong (1967) Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.
  • V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups A k , D k , E k and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
  • V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
  • V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
  • 10: 36.11 Leading-Order Asymptotics
    With real critical points (36.4.1) ordered so that …
    Asymptotics along Symmetry Lines