# Zero-free regions

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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.

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
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.
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
Mar 11 2018 0.4 0.4 $20 \leq N \leq 150$ ($5026 \leq x \leq 2.87 \times 10^5$) Rudolph & [1] Mesh evaluation of $A^{eff}+B^{eff} / B^{eff}_0$ and upper bounds on error terms