# critical strip

The of the Riemann zeta function is $\{s\in\mathbb{C}:0\leq\operatorname{Re}(s)\leq 1\}$. 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; about these zeroes is crucial in analytic number theory and the 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 (http://planetmath.org/RiemannZetaFunction).

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 (http://planetmath.org/RiemannZetaFunctionHasNoZerosOnReS01) for a proof.) Therefore, some people use the “critical strip” to refer to the region (http://planetmath.org/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 in the title of the entry formulae for zeta in the critical strip.

Title critical strip CriticalStrip 2013-03-22 16:07:21 2013-03-22 16:07:21 Wkbj79 (1863) Wkbj79 (1863) 11 Wkbj79 (1863) Definition msc 11M06 FormulaeForZetaInTheCriticalStrip ValueOfTheRiemannZetaFunctionAtS0 AnalyticContinuationOfRiemannZeta