The Riemann hypothesis is true up to 3·1012

  title={The Riemann hypothesis is true up to 3·1012},
  author={Dave Platt and Tim Trudgian},
  journal={Bulletin of the London Mathematical Society},
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 
On the error term in the explicit formula of Riemann--von Mangoldt
We provide an explicit $O(x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (2016) explicit. We also include applications to primes between
Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form
. We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.
Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with U (1) c symmetry. From this
A Proof to the Riemann Hypothesis Using a Simplified Xi-Function
: The Riemann hypothesis has been of great interest in the mathematics community since it was proposed by Bernhard Riemann in 1859, and makes important implications about the distribution of prime
Density results for the zeros of zeta applied to the error term in the prime number theorem
. We improve the unconditional explicit bounds for the error term in the prime counting function ψ ( x ). In particular, we prove that, for all x > 2, we have | ψ ( x ) − x | < 9 . 22106 x (log x ) 3
A Dynamical Systems Framework for Generating the Riemann Zeta Function and Dirichlet L-functions
Using an extension of the Price’s theorem we show how to construct a dynamical systems model which in its steady-state serves as an analytic continuation of the completed Riemann zeta function and
Amplitudes and the Riemann Zeta Function.
Physical properties of scattering amplitudes are mapped to the Riemann zeta function. Specifically, a closed-form amplitude is constructed, describing the tree-level exchange of a tower with masses
Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function
We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $\zeta(s)$. In particular, we provide the first unconditional
Price ’ s Theorem , the Riemann Zeta Function and the Riemann Hypothesis
Using an extension of the Price’s theorem we construct a dynamical systems model whose steady-state converges to the Riemann’s Xi function over a restricted domain that includes the critical strip.


New bounds for π(x)
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
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
Improvements to Turing's method II
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)
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
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
Estimating π(x) and related functions under partial RH assumptions
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