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critical strip
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(Definition)
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The critical strip of the Riemann zeta function is $\{ s \in \mathbb{C}: 0 \le \operatorname{Re}(s) \le 1 \}$ . The zeroes of the Riemann zeta function outside of the critical strip are exactly the set of all negative even integers. The location of the zeroes of the Riemann zeta function inside the critical strip is not totally known; information about these zeroes is crucial in analytic number theory and the distribution of primes. The Riemann hypothesis asserts that all zeroes of the Riemann zeta function that are in the critical strip lie on the line $\operatorname{Re}(s)=\frac{1}{2}$ . This is all explained in more detail in the entry Riemann zeta function.
It is well known that no zeroes of the Riemann zeta function lie on either of the lines $\operatorname{Re}(s)=0$ and $\operatorname{Re}(s)=1$ . (See this entry for a proof.) Therefore, some people use the term ``critical strip'' to refer to the region $\{ s \in \mathbb{C}: 0<\operatorname{Re}(s)<1 \}$ . (Note that this is the interior of the critical strip as defined above.) For example, this usage occurs in the title of the entry formulae for zeta in the critical strip.
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"critical strip" is owned by Wkbj79.
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Cross-references: formulae for zeta in the critical strip, interior, proof, line, lie on, Riemann hypothesis, primes, analytic number theory, even integers, negative, Riemann zeta function
There are 6 references to this entry.
This is version 8 of critical strip, born on 2006-07-29, modified 2008-03-15.
Object id is 8191, canonical name is CriticalStrip.
Accessed 2253 times total.
Classification:
| AMS MSC: | 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $) |
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Pending Errata and Addenda
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