The critical line is the line ![Re[s]=1/2](/megan-https-mathworld.wolfram.com/images/equations/CriticalLine/Inline1.svg)
in the complex plane
on which the Riemann hypothesis asserts that
all nontrivial zeros of the Riemann
zeta function lie. The plot above shows the first few zeros
of the Riemann zeta function, with the critical
line shown in red. The zeros with ![Im[s]=0](/megan-https-mathworld.wolfram.com/images/equations/CriticalLine/Inline2.svg)
and ![Re[s]<0](/megan-https-mathworld.wolfram.com/images/equations/CriticalLine/Inline3.svg)
that do not lie on the critical line are the trivial
zeros of 
at 
, 
, .... An infinite number of
zeros lie on the critical line. Pratt et al. (2020)
proved that more than 
of the nontrivial zeros, counted with multiplicity,
lie on it. Here a proportion refers to the limiting lower proportion among zeros
of positive imaginary part at most 
as 
.
Alpöge and Furman (2026) announced the stronger proportion 0.6725 for zeros of multiplicity 1 on the critical line. Their work credits an autonomous Claude run with discovering the proof and includes a Lean formalization. This claim is distinct from the published bound above. Neither bound resolves the Riemann hypothesis.
An attractive poster plotting the Riemann zeta function zeros on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).
In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes discusses the critical line after realizing that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security.
Are on the Critical Line." Res. Math. Sci. 7,
Art. 2, 2020. 