The Riemann hypothesis is true up to 3·1012
@article{Platt2021TheRH, title={The Riemann hypothesis is true up to 3·1012}, author={Dave Platt and Tim Trudgian}, journal={Bulletin of the London Mathematical Society}, year={2021}, volume={53} }
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3·1012 . That is, all zeroes β+iγ of the Riemann zeta‐function with 0
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References
SHOWING 1-10 OF 49 REFERENCES
New bounds for π(x)
- MathematicsMath. Comput.
- 2015
The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.
The Riemann Hypothesis
- Mathematics
- 2013
In this article I describe a proof of the fact that ZFC cannot decide whether a certain modified Turing machine, or computer (satisfying a certain condition) will ever halt successfully in finite…
Explicit zero density for the Riemann zeta function
- MathematicsJournal of Mathematical Analysis and Applications
- 2018
Improvements to Turing's method II
- Mathematics
- 2016
This article improves the estimate of the size of the definite inte- gral of S(t), the argument of the Riemann zeta-function. The primary appli- cation of this improvement is Turing's Method for the…
Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)
- Mathematics
- 2013
We prove that the error term $\sum_{n\le x} \Lambda(n)/n − \log x + \gamma$ differs from $(\psi(x) − x)/x$ by a well controlled function. We deduce very precise numerical results from this formula.
Explicit estimates of some functions over primes
- Mathematics
- 2018
New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann…
An improved upper bound for the argument of the Riemann zeta-function on the critical line II
- Mathematics
- 2012
Estimating π(x) and related functions under partial RH assumptions
- MathematicsMath. Comput.
- 2016
The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving…