# Improvements to Turing's method II

@article{Trudgian2016ImprovementsTT,
title={Improvements to Turing's method II},
author={Tim Trudgian},
journal={Rocky Mountain Journal of Mathematics},
year={2016},
volume={46},
pages={325-332}
}
• T. Trudgian
• Published 13 June 2014
• Mathematics
• Rocky Mountain Journal of Mathematics
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 Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet L-functions and of Dedekind zeta-functions.
10 Citations
Numerical computations concerning the GRH
• Dave Platt
• Computer Science, Mathematics
• Math. Comput.
• 2016
We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated withExpand
On explicit estimates for S(t), S1(t), and ζ(1/2+it) under the Riemann Hypothesis
Abstract Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of S ( t ) , S 1 ( t ) , and ζ ( 1 / 2 + i t ) while comparing them with recently proven unconditional ones. As aExpand
The Riemann hypothesis is true up to $3\cdot 10^{12}$
• Mathematics
• 2020
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $\beta + i\gamma$ of the RiemannExpand
A pr 2 02 0 The Riemann hypothesis is true up to 3 · 10 12
• 2020
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3 · 10. That is, all zeroes β + iγ of the Riemann zeta-function with 0 < γ ≤ 3 ·Expand
Explicit zero density estimate for the Riemann zeta-function near the critical line
• A. Simonič
• Mathematics
• Journal of Mathematical Analysis and Applications
• 2020
In 1946, A. Selberg proved $N(\sigma,T) \ll T^{1-\frac{1}{4} \left(\sigma-\frac{1}{2}\right)} \log{T}$ where $N(\sigma,T)$ is the number of nontrivial zeros $\rho$ of the Riemann zeta-function withExpand
An improved explicit bound on $|\zeta(1/2 + it)|$
• Mathematics
• 2014
This article proves the bound $|\zeta(\frac{1}{2} + it)|\leq 0.732 t^{\frac{1}{6}} \log t$ for $t \geq 2$, which improves on a result by Cheng and Graham. We also show thatExpand
Accurate estimation of sums over zeros of the Riemann zeta-function
• Computer Science, Mathematics
• Math. Comput.
• 2021
We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. WeExpand