Difference between revisions of "Zero-free regions"

From Polymath1Wiki
Jump to: navigation, search
(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...")
 
Line 38: Line 38:
 
| [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
 +
|
 
|-
 
|-
 
| Mar 11 2018
 
| Mar 11 2018
Line 45: Line 46:
 
| [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
|  
+
| Should extend to the range <math>0.4 \leq y \leq 0.45</math>
 
|-
 
|-
 
| Mar 11 2018
 
| Mar 11 2018
Line 51: Line 52:
 
| 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
 
| 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>)
 +
| [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
 +
|
 +
|-
 +
| 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>
 +
|
 +
|-
 +
| 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>
 +
|
 +
|-
 +
| 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
 +
|
 +
|-
 +
| 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>
 +
|
 +
|-
 +
| 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>!
 
|}
 
|}

Revision as of 10:42, 29 March 2018

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.


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