Zero-free regions: Difference between revisions

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Created page with "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..."
 
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| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493734 KM]
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493734 KM]
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
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| Mar 11 2018
| Mar 11 2018
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| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493762 KM]
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493762 KM]
| Analytic lower bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
| Analytic lower bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
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| Should extend to the range <math>0.4 \leq y \leq 0.45</math>
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| Mar 11 2018
| Mar 11 2018
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| 0.4
| 0.4
| <math>20 \leq N \leq 150</math> (<math>5026 \leq x \leq 2.87 \times 10^5</math>)
| <math>20 \leq N \leq 150</math> (<math>5026 \leq x \leq 2.87 \times 10^5</math>)
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493769 Rudolph] & [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493771]
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493769 Rudolph] & [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493771 KM]
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
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| Mar 11 2018
| 0.4
| 0.4
| <math>11 \leq N \leq 19</math> (<math>1520 \leq x \leq 5026</math>)
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493769 Rudolph] & [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493771 KM]
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
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| Mar 22 2018
| 0.4
| 0.4
| <math>x \leq 1000</math>
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-494166 Anon/David/KM]
| Mesh evaluation of <math>H_t</math>
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| Mar 22 2018
| 0.4
| 0.4
| <math>1000 \leq x \leq 1600</math>
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-494175 Rudolph]
| Mesh evaluation of <math>H_t</math>
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| Mar 22 2018
| 0.4
| 0.4
| <math>8 \leq N \leq 10</math> (<math>803 \leq x \leq 1520</math>)
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-494175 Rudolph]
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
|
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| Mar 23 2018
| 0.4
| 0.4
| <math>20 \leq x \leq 1000</math>
| [https://terrytao.wordpress.com/2018/03/18/polymath15-sixth-thread-the-test-problem-and-beyond/#comment-494238 Anonymous]
| Mesh evaluation of <math>H_t</math>
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| 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>)
| [https://terrytao.wordpress.com/2018/03/18/polymath15-sixth-thread-the-test-problem-and-beyond/#comment-494859 KM]
| Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
|
|-
| Mar 27 2018
| 0.4
| <math>0.4 \leq y \leq 0.45</math>
| <math>0 \leq x \leq 1000</math>
| [https://terrytao.wordpress.com/2018/03/18/polymath15-sixth-thread-the-test-problem-and-beyond/#comment-494867 Anonymous]
| Mesh evaluation of <math>H_t</math>
| Completes proof of <math>\Lambda \leq 0.48</math>!
|}
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Revision as of 09:42, 29 March 2018

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.


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
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.
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 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
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] Completes proof of [math]\displaystyle{ \Lambda \leq 0.48 }[/math]!
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