Classical zero-free region constant
Description of constant
$C_{8} = R$ is the least constant such that there are no zeroes $\sigma+it$ of the Riemann zeta function with $\lvert t \rvert \geq 2$ and $\sigma > 1 - \frac{1}{R \log \lvert t \rvert}$.
Known upper bounds
| Bound |
Reference |
Comments |
| $0$ |
Trivial |
|
| $34.82$ |
[dlVP1899] |
Implies the prime number theorem |
| $19$ |
[RS1962] |
|
| $9.64591$ |
[S1970] |
|
| $8.463$ |
[F2002] |
|
| $5.69693$ |
[K2005] |
|
| $5.68371$ |
[JK2014] |
|
| $5.5666305$ |
[MT2014] |
|
| $5.558691$ |
[MTY2022] |
|
Known lower bounds
| Bound |
Reference |
Comments |
| $2/\log \gamma_{1} \approx 0.755106$ |
- |
Optimal assuming RH |
- This constant is relevant to the classical error term in the prime number theorem; in particular, $\pi(x) = \mathrm{Li}(x) + O\left(x \exp\left(-\sqrt{\log x / R}\right)\right)$.
References
- [F2002] Ford, Kevin. Vinogradov’s integral and bounds for the Riemann zeta function. Proceedings of the London Mathematical Society, 85(3):565-633, 2002.
- [F2002b] Ford, Kevin. Zero-free regions for the Riemann zeta function. In Number Theory for the Millennium, II (Urbana, IL, 2000), pages 25-56. A K Peters, 2002.
- [JK2014] Jang, Won-Jin; Kwon, Seunghyun. A note on Kadiri’s explicit zero free region for Riemann zeta function. Journal of the Korean Mathematical Society, 51(6):1291-1304, 2014.
- [K2005] Kadiri, Habiba. Une région explicite sans zéros pour la fonction ζ de Riemann. Acta Arithmetica, 117(4):303-339, 2005.
- [K1977] Kondrat’ev, V. P. Some extremal properties of positive trigonometric polynomials. Mathematical notes of the Academy of Sciences of the USSR, 22(3):696-698, 1977.
- [MT2014] Mossinghoff, Michael J.; Trudgian, Timothy S. Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function. Journal of Number Theory, 157:329-349, 2015.
- [MTY2022] Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew. Explicit zero-free regions for the Riemann zeta-function. arXiv preprint arXiv:2212.06867, 2022.
- [R1941] Rosser, J. Barkley. Explicit bounds for some functions of prime numbers. American Journal of Mathematics, 63(2):211-232, 1941.
- [RS1962] Rosser, J. Barkley; Schoenfeld, Lowell. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1):64-94, 1962.
- [RS1975] Rosser, J. Barkley; Schoenfeld, Lowell. Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Mathematics of Computation, 29(129):243-269, 1975.
- [S1970] Stechkin, S. B. Zeros of the Riemann zeta-function. Mathematical notes of the Academy of Sciences of the USSR, 8(4):706-711, 1970.
-[dlVP1899] de la Vallée Poussin, Charles-Jean. Sur la fonction ζ (s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mémoires de l’Académie royale de Belgique, 59:1-74, 1899.