For a complex number s with operatornameRe(s)>1, let zeta(s)=∑_n=1^∞frac1n^s, the series being absolutely convergent, and let zeta also denote its meromorphic continuation to C, which is holomorphic except for a simple pole at s=1 with residue 1 and satisfies zeta(s)=2^spi^s-1sin!(fracpi s2)Gamma(1-s),zeta(1-s) for all s∈Csetminus1. The zeros of zeta at s=-2,-4,-6,… are called trivial; all remaining zeros are called nontrivial, and every nontrivial zero satisfies 0<operatornameRe(s)<1. Every nontrivial zero of zeta has real part tfrac12; that is, for every s∈C, zeta(s)=0 and 0<operatornameRe(s)<1 quadLongrightarrow quad operatornameRe(s)=tfrac12 .

Riemann hypothesis

For a complex number $s$ with $\operatorname{Re}(s)>1$ , let

\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},

the series being absolutely convergent, and let $\zeta$ also denote its meromorphic continuation to $\mathbb{C}$ , which is holomorphic except for a simple pole at $s=1$ with residue $1$ and satisfies

\zeta(s)=2^{s}\pi^{s-1}\sin\!\left(\frac{\pi s}{2}\right)\Gamma(1-s)\,\zeta(1-s)

for all $s\in\mathbb{C}\setminus\{1\}$ . The zeros of $\zeta$ at $s=-2,-4,-6,\dots$ are called trivial; all remaining zeros are called nontrivial, and every nontrivial zero satisfies $0<\operatorname{Re}(s)<1$ .

Every nontrivial zero of $\zeta$ has real part $\tfrac12$ ; that is, for every $s\in\mathbb{C}$ ,

\zeta(s)=0\ \text{ and }\ 0<\operatorname{Re}(s)<1\quad\Longrightarrow\quad \operatorname{Re}(s)=\tfrac12 .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Riemann's hypothesis
For $s\in\mathbb{C}$ , define the Riemann zeta function by
$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$
It converges for $\operatorname{Re}(s)>1$ . Riemann hypothesis. All non-trivial zeros of $\zeta(s)$ lie on the line $\operatorname{Re}(s)=\frac{1}{2}$ . The hypothesis is one of the central unsolved problems in number theory and has deep consequences for the distribution of prime numbers.
source: Madhuparna Das, “Mapping Mathematical Hardness: Machine-Assisted Conjecture Discovery and the Quantification of Non-Triviality”, arXiv:2606.14804 (2026).

References

Primary source

Clay Mathematics Institute

Additional references

Enrico Bombieri, Problems of the Millennium: The Riemann Hypothesis, the official Clay Mathematics Institute problem description.
B. Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," Monatsberichte der Berliner Akademie (1859).
H. M. Edwards, Riemann's Zeta Function, Academic Press (1974).
H. von Koch, "Sur la distribution des nombres premiers," Acta Mathematica 24 (1901), 159-182 — the equivalence with the error term in the prime number theorem.
Clay Mathematics Institute, Riemann hypothesis — the statement above follows it.
Wikipedia, Riemann hypothesis, the article this problem comes from.

unsolved-problem

InMillennium Prize Problems

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

The hypothesis remains unproved, although one manuscript claims a complete proof and other recent work has established stronger partial results.

Proposed by Bernhard Riemann in 1859, the hypothesis says that every nontrivial zero of the zeta function lies on one central vertical line.

Known results

Hardy (1914) proved that infinitely many zeros lie on $\operatorname{Re}(s)=\tfrac12$ .
Selberg, Levinson, Conrey, and Feng established progressively stronger lower bounds, reaching at least $41.28\%$ of zeros on the critical line.
Guth and Maynard (2024) improved a zero-density exponent from $\tfrac35$ to $\tfrac{13}{25}$ .
Computation has verified zeros through height $3\cdot10^{12}$ , but only finitely many.

September 2026 claimed proof and partial advances

On September 7, 2026, I. Murtazin announced a manuscript claiming a complete proof; this claim is unverified. August 2026 work attributed to Anthropic claimed that more than $67.2\%$ of zeros lie on the critical line, without settling the hypothesis. On September 22, Mishra and Sarkar gave an equivalent finite arithmetic formulation, while leaving its universal assertion open.

Current status (as of September 2026): The hypothesis remains open; Murtazin's claimed proof is unverified, while the Anthropic and finite-formulation results are partial.

Claude (unreleased research version)Anthropicpartial progress2026-08-16evidence
Anthropic reports a 67.2% critical-line proportion for zeta zeros

Sources

No solutions have been posted yet.

Riemann's hypothesis

Primary source

Additional references

Progress summary