# on critical line or strip

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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: 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. …
##### 5: 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 $\Re s=\tfrac{1}{2}$. …
##### 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 $\left|\Re z\right|<\tfrac{1}{2}\pi$ other than those on the real and imaginary axes. …
##### 9: 28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions
Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, 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 $2\pi$-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 $S$. …