Zero-free regions

The table below lists various regions of the [math]\displaystyle{ (t,y,x) }[/math] parameter space where [math]\displaystyle{ H_t(x+iy) }[/math] is known to be non-zero. In some cases the parameter

[math]\displaystyle{ N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor }[/math]

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