Can MathGPT solve the Riemann Hypothesis?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

UnsolvedNumber theoryMillennium Prize Problem

Riemann Hypothesis

Q: Is there a prize?

Prize of US \$1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

Every non-trivial zero of the Riemann zeta function has real part 1/2.

Exact statement

\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\quad(\operatorname{Re}s>1),\ \text{continued analytically to }\mathbb{C}\setminus\{1\}

\zeta(\rho)=0,\ 0<\operatorname{Re}\rho<1\ \Longrightarrow\ \operatorname{Re}\rho=\tfrac12

All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. (The trivial zeros are the negative even integers −2, −4, −6, ….)

Status

Open. Proposed by Bernhard Riemann in 1859; no proof or counterexample is known.

Posed: 1859, Bernhard Riemann.

Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

In plain words

The zeta function packs the prime numbers into one function: Euler showed that $\zeta(s)=\prod_p (1-p^{-s})^{-1}$ over all primes $p$ . Riemann discovered that the exact distribution of the primes is controlled by where $\zeta$ is zero.

Apart from the 'trivial' zeros at $-2,-4,-6,\dots$ , every zero lies in the strip $0<\operatorname{Re}s<1$ . The hypothesis says they all sit exactly on the middle line $\operatorname{Re}s=\tfrac12$ . The first one is about $\tfrac12+14.134725\,i$ .

If true, the prime-counting function $\pi(x)$ stays within about $\sqrt{x}\log x$ of the logarithmic integral $\operatorname{Li}(x)$ (von Koch, 1901): the primes are as evenly spread as they possibly could be. Hundreds of theorems are proved 'assuming RH'.

Key results so far

1896: Hadamard and de la Vallée Poussin prove there are no zeros on the line Re(s) = 1, which gives the Prime Number Theorem.
1914: Hardy proves infinitely many zeros lie on the critical line.
1942: Selberg proves a positive proportion of zeros lie on the line; Levinson (1974) raises it to more than one third, Conrey (1989) to more than two fifths.
2021: Platt and Trudgian verify by rigorous computation that every zero with imaginary part up to 3·10¹² lies on the critical line.

Verified zeros and exact values

Verified zeros

The first 10¹³ zeros have been computed numerically on the line (Gourdon, 2004), and rigorously up to height 3·10¹² (Platt and Trudgian, 2021). Evidence, not proof: the statement is about infinitely many zeros.

Known values

\zeta(2)=\frac{\pi^2}{6},\quad \zeta(-1)=-\frac{1}{12},\quad \zeta(-2n)=0

Exact values from Euler and the functional equation.

What our solver can and cannot do

No computer program or AI can solve the Riemann Hypothesis by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Riemann Hypothesis say?: All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. (The trivial zeros are the negative even integers −2, −4, −6, ….)
Has the Riemann Hypothesis been solved?: Open. Proposed by Bernhard Riemann in 1859; no proof or counterexample is known.
Is there a prize?: Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).
Can MathGPT solve the Riemann Hypothesis?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

E. Bombieri, 'Problems of the Millennium: the Riemann Hypothesis', Clay Mathematics Institute (2000).
H. M. Edwards, Riemann's Zeta Function, Academic Press (1974).
D. Platt and T. Trudgian, 'The Riemann hypothesis is true up to 3·10¹²', Bull. London Math. Soc. 53 (2021).

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