Zero-free regions

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

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

is used instead of [math]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]t[/math] [math]y[/math] [math]x[/math] From Method Comments
1950 [math]t \geq 0[/math] [math]y \gt \sqrt{\max(1-2t,0)}[/math] Any De Bruijn Theorem 13 of de Bruijn Proves [math]\Lambda \leq 1/2[/math].
2004 0 [math]y\gt0[/math] [math]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]t \gt 0[/math] [math]y \gt 0[/math] [math]x \geq C(t)[/math] Ki-Kim-Lee Theorem 1.3 of Ki-Kim-Lee [math]C(t)[/math] is not given explicitly. Also they show [math]\Lambda \lt 1/2[/math].
2017 0 [math]y\gt0[/math] [math]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]N \geq 2000[/math] ([math]x \geq 5.03 \times 10^7[/math]) Tao Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and analytic upper bounds on error terms Can be extended to the range [math]0.4 \leq y \leq 0.45[/math]
Mar 10 2018 0.4 0.4 [math]151 \leq N \leq 300[/math] ([math]2.87 \times 10^5 \leq x \leq 1.13 \times 10^6[/math]) KM Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 11 2018 0.4 0.4 [math]300 \leq N \leq 2000[/math] ([math]1.13 \times 10^6 \leq x \leq 5.03 \times 10^7[/math]) KM Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms Should extend to the range [math]0.4 \leq y \leq 0.45[/math]
Mar 11 2018 0.4 0.4 [math]20 \leq N \leq 150[/math] ([math]5026 \leq x \leq 2.87 \times 10^5[/math]) Rudolph & KM Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 11 2018 0.4 0.4 [math]11 \leq N \leq 19[/math] ([math]1520 \leq x \leq 5026[/math]) Rudolph & KM Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 22 2018 0.4 0.4 [math]x \leq 1000[/math] Anon/David/KM Mesh evaluation of [math]H_t[/math]
Mar 22 2018 0.4 0.4 [math]1000 \leq x \leq 1600[/math] Rudolph Mesh evaluation of [math]H_t[/math]
Mar 22 2018 0.4 0.4 [math]8 \leq N \leq 10[/math] ([math]803 \leq x \leq 1520[/math]) Rudolph Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms
Mar 23 2018 0.4 0.4 [math]20 \leq x \leq 1000[/math] Anonymous Mesh evaluation of [math]H_t[/math]
Mar 23 2018 [math]t \gt 0[/math] [math]y \gt 0[/math] [math]x \gt \exp(C/t)[/math] Tao Analytic bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and error terms; argument principle [math]C[/math] is in principle an explicit absolute constant
Mar 27 2018 0.4 [math]0.4 \leq y \leq 0.45[/math] [math]7 \leq N \leq 300[/math] ([math]615 \leq x \leq 1.13 \times 10^6[/math]) KM Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Mar 27 2018 0.4 [math]0.4 \leq y \leq 0.45[/math] [math]0 \leq x \leq 1000[/math] Anonymous Mesh evaluation of [math]H_t[/math]; argument principle Completes proof of [math]\Lambda \leq 0.48[/math]!
Mar 31 2018 [math]0 \leq t \leq 0.4[/math] [math]0.4 \leq y \leq 1[/math] [math]10^6 \leq x \leq 10^6 + 1 [/math] KM Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Mar 31 2018 0.4 [math]0.4 \leq y \leq 0.45[/math] [math]0 \leq x \leq 3000[/math] Rudolph Third approach to [math]H_t[/math]; argument principle
Apr 6 2018 [math]0 \leq t \leq 0.2[/math] [math]0.4 \leq y \leq 1[/math] [math]5 \times 10^9 \leq x \leq 5 \times 10^9+1[/math] KM, Rudolph, David, Anonymous Mesh evaluation of [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms; argument principle
Apr 6 2018 0.2 [math]y = 0.4[/math] [math]N \geq 300000 (x \geq 1.13 \times 10^{12})[/math] KM Analytic lower bounds on [math]A^{eff}+B^{eff} / B^{eff}_0[/math] and upper bounds on error terms