critical strip

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1: 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 $\mathrm{\Re }s=\frac{1}{2}$. … ►Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta \left(s\right)$ in that portion of the critical strip with . … ►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)). …

2: Bibliography V

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On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt

Referenced by:

§25.10(ii), §25.18(ii).

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3: 25.18 Methods of Computation

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§25.18(ii) Zeros

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4: 25.15 Dirichlet $L$-functions

… ►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. …

5: Errata

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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).

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