# critical line

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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 $\zeta\left(\frac{1}{2}+it\right)$ 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 $\Re s=\frac{1}{2}$. Calculations to date (2008) have found no nontrivial zeros off the critical line. …
##### 2: 25.10 Zeros
###### §25.10(i) Distribution
In the region $0<\Re s<1$, called the critical strip, $\zeta\left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\Re s=\frac{1}{2}$. … 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
Analogies exist between the distribution of the zeros of $\zeta\left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. …
##### 4: 8.22 Mathematical Applications
The function $\Gamma\left(a,z\right)$, with $|\operatorname{ph}a|\leq\tfrac{1}{2}\pi$ and $\operatorname{ph}z=\tfrac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta\left(s\right)$25.2(i)) on the critical line $\Re s=\tfrac{1}{2}$. …
##### 5: 25.15 Dirichlet $L$-functions
There are also infinitely many zeros in the critical strip $0\leq\Re s\leq 1$, located symmetrically about the critical line $\Re s=\frac{1}{2}$, 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: 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).
• ##### 9: 31.15 Stieltjes Polynomials
The zeros $z_{k}$, $k=1,2,\ldots,n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\ifrac{\partial G}{\partial\zeta_{k}=0}$, $k=1,2,\ldots,n$, where …