# critical line

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## 1—10 of 13 matching pages

##### 1: 25.18 Methods of Computation

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###### §25.18(ii) Zeros

►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$ in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of $\zeta \left(s\right)$ lie on the critical line $\mathrm{\Re}s=\frac{1}{2}$. Calculations to date (2008) have found no nontrivial zeros off the critical line. …##### 2: 25.10 Zeros

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###### §25.10(i) Distribution

… ►In the region $$, called the*critical strip*, $\zeta \left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the*critical line*$\mathrm{\Re}s=\frac{1}{2}$. … ►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 $\zeta \left(s\right)$ 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

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►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues.
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##### 4: 8.22 Mathematical Applications

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►The function $\mathrm{\Gamma}(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi $ and $\mathrm{ph}z=\frac{1}{2}\pi $, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line
$\mathrm{\Re}s=\frac{1}{2}$.
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##### 5: 25.15 Dirichlet $L$-functions

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►There are also infinitely many zeros in the critical strip $0\le \mathrm{\Re}s\le 1$, located symmetrically about the critical line
$\mathrm{\Re}s=\frac{1}{2}$, but not necessarily symmetrically about the real axis.
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##### 6: Bibliography B

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More than 41% of the zeros of the zeta function are on the critical line.
Acta Arith. 150 (1), pp. 35–64.
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##### 7: Errata

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Subsection 25.10(ii)
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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

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►The zeros ${z}_{k}$, $k=1,2,\mathrm{\dots},n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\partial G/\partial {\zeta}_{k}=0$, $k=1,2,\mathrm{\dots},n$, where
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##### 9: Bibliography

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Evaluation of Coulomb wave functions along the transition line.
Physical Rev. (2) 96, pp. 77–79.
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Spectrum line profiles: The Voigt function.
J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.
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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).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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