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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: 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. …
5: 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 . …
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: 28.9 Zeros
    There are no zeros within the strip | z | < 1 2 π other than those on the real and imaginary axes. …
    9: 28.19 Expansions in Series of me ν + 2 n Functions
    Let q be a normal value (§28.12(i)) with respect to ν , and f ( z ) be a function that is analytic on a doubly-infinite open strip S that contains the real axis. …
    10: 28.11 Expansions in Series of Mathieu Functions
    Let f ( z ) be a 2 π -periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value (§28.7). …The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip S . …