# Difference between revisions of "Zero-free regions"

The table below lists various regions of the $(t,y,x)$ parameter space where $H_t(x+iy)$ is known to be non-zero. In some cases the parameter

$N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor$

is used instead of $x$. The mesh evaluation techniques also require rigorous upper bounds on derivatives. In some cases the spacing of the mesh is fixed; in other cases it is adaptive based on the current value of the evaluation and on the derivative bound.

Date $t$ $y$ $x$ From Method Comments
1950 $t \geq 0$ $y \gt \sqrt{\max(1-2t,0)}$ Any De Bruijn Theorem 13 of de Bruijn Proves $\Lambda \leq 1/2$.
2004 0 $y\gt0$ $0 \leq x \leq 4.95 \times 10^{11}$ Gourdon-Demichel Numerical verification of RH & Riemann-von Mangoldt formula Results have not been independently verified
2009 $t \gt 0$ $y \gt 0$ $x \geq C(t)$ Ki-Kim-Lee Theorem 1.3 of Ki-Kim-Lee $C(t)$ is not given explicitly. Also they show $\Lambda \lt 1/2$.
2017 0 $y\gt0$ $0 \leq x \leq 6.1 \times 10^{10}$ Platt Numerical verification of the Riemann hypothesis
Mar 7 2018 0.4 0.4 $N \geq 2000$ ($x \geq 5.03 \times 10^7$) Tao Analytic lower bounds on $A^{eff}+B^{eff} / B^{eff}_0$ and analytic upper bounds on error terms Can be extended to the range $0.4 \leq y \leq 0.45$
Mar 10 2018 0.4 0.4 $151 \leq N \leq 300$ ($2.87 \times 10^5 \leq x \leq 1.13 \times 10^6$) KM Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Mar 11 2018 0.4 0.4 $300 \leq N \leq 2000$ ($1.13 \times 10^6 \leq x \leq 5.03 \times 10^7$) KM Analytic lower bounds on $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms Should extend to the range $0.4 \leq y \leq 0.45$
Mar 11 2018 0.4 0.4 $20 \leq N \leq 150$ ($5026 \leq x \leq 2.87 \times 10^5$) Rudolph & KM Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Mar 11 2018 0.4 0.4 $11 \leq N \leq 19$ ($1520 \leq x \leq 5026$) Rudolph & KM Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Mar 22 2018 0.4 0.4 $x \leq 1000$ Anon/David/KM Mesh evaluation of $H_t$
Mar 22 2018 0.4 0.4 $1000 \leq x \leq 1600$ Rudolph Mesh evaluation of $H_t$
Mar 22 2018 0.4 0.4 $8 \leq N \leq 10$ ($803 \leq x \leq 1520$) Rudolph Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Mar 23 2018 0.4 0.4 $20 \leq x \leq 1000$ Anonymous Mesh evaluation of $H_t$
Mar 23 2018 $t \gt 0$ $y \gt 0$ $x \gt \exp(C/t)$ Tao Analytic bounds on $A^{eff}+B^{eff} / B^{eff}_0$ and error terms; argument principle $C$ is in principle an explicit absolute constant
Mar 27 2018 0.4 $0.4 \leq y \leq 0.45$ $7 \leq N \leq 300$ ($615 \leq x \leq 1.13 \times 10^6$) KM Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms; argument principle
Mar 27 2018 0.4 $0.4 \leq y \leq 0.45$ $0 \leq x \leq 1000$ Anonymous Mesh evaluation of $H_t$; argument principle Completes proof of $\Lambda \leq 0.48$!
Mar 31 2018 $0 \leq t \leq 0.4$ $0.4 \leq y \leq 1$ $10^6 \leq x \leq 10^6 + 1$ KM Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms; argument principle
Mar 31 2018 0.4 $0.4 \leq y \leq 0.45$ $0 \leq x \leq 3000$ Rudolph Third approach to $H_t$; argument principle
Apr 6 2018 $0 \leq t \leq 0.2$ $0.4 \leq y \leq 1$ $5 \times 10^9 \leq x \leq 5 \times 10^9+1$ KM, Rudolph, David, Anonymous Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms; argument principle
Apr 6 2018 0.2 0.4 $N \geq 3 \times 10^6 (x \geq 1.13 \times 10^{12})$ KM Analytic lower bounds on $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Apr 7 2018 0.29 $y \geq 0.29$ $N \geq 19947$ Anonymous Triangle inequality bound on $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms Would in principle show $\Lambda \leq 0.33205$ if the matching barrier could be established
Apr 9 2018 0.2 $y \geq 0.4$ $N \geq 3 \times 10^5$ Tao Triangle inequality bound on $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms
Apr 10 2018 0.2 $y \geq 0.4$ $4 \times 10^4 \leq N \leq 10^5; 100|N$ KM Euler2 mollifier and triangle inequality bounds on $A^{eff}+B^{eff} / B^{eff}_0$ Error terms not estimated but look well within acceptable limits
Apr 12 2018 0.2 $y \geq 0.4$ $4 \times 10^4 \leq N \leq 3 \times 10^5$ Anonymous Euler2 mollifier and triangle inequality bounds on $A^{eff}+B^{eff} / B^{eff}_0$ Error terms not estimated but look well within acceptable limits
Apr 12 2018 0.2 $y \geq 0.4$ $19947 \leq N \leq 4 \times 10^4$ Rudolph Euler3 mollifier and triangle inequality bounds on $A^{eff}+B^{eff} / B^{eff}_0$ Error terms not estimated but look well within acceptable limits
Apr 16 2018 0.2 $y \geq 0.4$ $19947 \leq N \leq 3 \times 10^5$ Rudolph Estimating error terms in previous two ranges Completes proof of $\Lambda \leq 0.28$!
Apr 28 2018 $0 \leq t \leq 0.2$ $0.2 \leq y \leq 1$ $6 \times 10^{10} + 2099 \leq x \leq 6 \times 10^{10} + 2100$ KM / Rudolph Taylor series evaluation of $A^{eff}+B^{eff} / B^{eff}_0$
May 1 2018 $0 \leq t \leq 0.2$ $0.2 \leq y \leq 1$ $6 \times 10^{10} + 83952 - 1/2\leq x \leq 6 \times 10^{10} + 83952 + 1/2$ KM Taylor series evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ Barrier chosen using Euler product heuristics to maximise lower bound
May 1 2018 0.2 0.2 $69098 \leq N \leq 1.5 \times 10^6$ Rudolph Euler5 mollifier and triangle inequality bounds on $A^{eff}+B^{eff} / B^{eff}_0$; only updating a few terms at a time when moving from $N$ to $N+1$ This should establish $\Lambda \leq 0.22$!