Open problem, Arithmetic and number theory, Analysis, posed 1859

Riemann hypothesis

OpenMillenniumHilbert #8

All nontrivial zeros of the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ (extended by analytic continuation to $\mathbb{C}\setminus\{1\}$ ) have real part exactly $\tfrac{1}{2}$ .

Posed by, 1859Bernhard Riemann

Research frontier as of 2026

As of 2026 the Riemann hypothesis remains open. The strongest unconditional zero-density estimate comes from Guth and Maynard's 2024 improvement of Ingham's 1940 bound, giving $N(\sigma,T) \le T^{\frac{30(1-\sigma)}{13}+o(1)}$ (arXiv:2405.20552); the paper is forthcoming in the Annals of Mathematics. This is the first structural improvement to the exponent in over eighty years, and it sharpens results on primes in short intervals. More than 5/12 of all nontrivial zeros are known unconditionally to lie on the critical line, and over 40.75% are known to be simple and on the critical line (Pratt–Robles–Zaharescu–Zeindler, 2020). Numerically, the first $10^{13}$ zeros (Gourdon, 2004) and large additional blocks near height $10^{24}$ all lie on the line, but no finite computation can settle an infinite statement.

Best known results

Guth–Maynard (2024) zero-density bound $N(\sigma,T) \le T^{\frac{30(1-\sigma)}{13}+o(1)}$ , the first structural improvement on Ingham's 1940 exponent in over 80 years (arXiv:2405.20552).
More than 5/12 of the nontrivial zeros are known to lie on the critical line, and over 40.75% are simple and on the critical line (Pratt–Robles–Zaharescu–Zeindler, 2020).
The first $10^{13}$ zeros, and large blocks of zeros near height $10^{24}$ , have been verified numerically to lie on the critical line (Gourdon, 2004).

Tools and where they stop

Tool	Achieved	Where it stops
Zero-density estimates for Dirichlet polynomials	Bounds how many zeros can lie off the critical line, giving unconditional results on primes in short intervals (Guth–Maynard 2024 push the exponent to 30/13).	Falls far short of ruling out zeros off the line entirely; it only shows they are rare.
Levinson's method (mollified moments)	Proves a positive proportion of zeros lie on the critical line (currently over 5/12, with over 40.75% simple, Pratt–Robles–Zaharescu–Zeindler 2020).	Cannot reach the full 100% needed for a proof, and improvements have slowed to a crawl.
Large-scale numerical verification	Confirms the first $10^{13}$ zeros, and spot-checks near height $10^{24}$ , lie exactly on the line (Gourdon, 2004).	Checking finitely many zeros can never prove a statement about all infinitely many of them.

Open questions

Can the zero-density exponent be pushed low enough to imply the Riemann hypothesis outright, or is a fundamentally different method needed?
Does the Guth–Maynard estimate give the optimal error term in the prime number theorem, or can the exponent $x^{17/30}$ for primes in short intervals be lowered further?

References

Bernhard Riemann (1859). Über die Anzahl der Primzahlen unter einer gegebenen Größe · DOI:10.1017/CBO9781139568050.008
Larry Guth, James Maynard (2024). New large value estimates for Dirichlet polynomials · arXiv:2405.20552 [preprint, not peer-reviewed]
Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler (2020). More than five-twelfths of the zeros of ζ are on the critical line · arXiv:1802.10521
Xavier Gourdon (2004). The $10^{13}$ first zeros of the Riemann zeta function, and zeros computation at very large height