Zero-free regions: Difference between revisions
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| Mesh evaluation of <math>H_t</math> | | Mesh evaluation of <math>H_t</math> | ||
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| Mar 23 2018 | |||
| <math>t > 0</math> | |||
| <math>y > 0</math> | |||
| <math>x > \exp(C/t)</math> | |||
| [https://terrytao.wordpress.com/2018/03/18/polymath15-sixth-thread-the-test-problem-and-beyond/#comment-494201 Tao] | |||
| Analytic of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and analytic bounds on error terms; argument principle | |||
| <math>C</math> is in principle an explicit absolute constant | |||
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| Mar 27 2018 | | Mar 27 2018 | ||
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| <math>7 \leq N \leq 300</math> (<math>615 \leq x \leq 1.13 \times 10^6</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] | | [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 | | Mesh evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms; argument principle | ||
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| <math>0 \leq x \leq 1000</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] | | [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> | | Mesh evaluation of <math>H_t</math>; argument principle | ||
| Completes proof of <math>\Lambda \leq 0.48</math>! | | Completes proof of <math>\Lambda \leq 0.48</math>! | ||
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Revision as of 09:48, 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 instead of [math]\displaystyle{ x }[/math].
| 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 23 2018 | [math]\displaystyle{ t \gt 0 }[/math] | [math]\displaystyle{ y \gt 0 }[/math] | [math]\displaystyle{ x \gt \exp(C/t) }[/math] | Tao | Analytic of [math]\displaystyle{ A^{eff}+B^{eff} / B^{eff}_0 }[/math] and analytic bounds on 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]! |
