Polymath15 grant acknowledgments

2018-12-30T18:35:53Z

Kmuchhal: Added section on grid computing contribution

Participants should be arranged in alphabetical order of surname.

=== Participants and contact information ===

Caution: this list may be incomplete. Participants who have made significant contributions to the project (on par with a co-author on a traditional mathematical research paper) should add themselves to this list, or email tao@math.ucla.edu if they are unable to do so directly. Participants who have made auxiliary contributions to the project (on par with those mentioned in an Acknowledgments section in a traditional paper) should add themselves instead to the list at the bottom of the page.

* Rudolph Dwars
* Kalpesh Muchhal, [https://kommonmann.wordpress.com/contact/]
* Terence Tao, UCLA, [http://www.math.ucla.edu/~tao]

=== Grid computing contributors ===

Some of the computations in the project were performed using a Boinc [https://boinc.berkeley.edu/] based grid computing setup at http://anthgrid.com/dbnupperbound
To see the participants and their contributions, please check http://anthgrid.com/dbnupperbound/top_users.php?sort_by=total_credit

=== Grant information ===

* Terence Tao was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

Zero-free regions

2018-05-07T08:00:00Z

Kmuchhal: Changed the X range slightly for the second last entry

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.

{| border=1
|-
!Date!!<math>t</math>!! <math>y</math> !! <math>x</math> !! From !! Method !! Comments
|-
| 1950
| <math>t \geq 0</math>
| <math>y > \sqrt{\max(1-2t,0)}</math>
| Any
| [https://pure.tue.nl/ws/files/1769368/597490.pdf De Bruijn]
| Theorem 13 of [https://pure.tue.nl/ws/files/1769368/597490.pdf de Bruijn]
| Proves <math>\Lambda \leq 1/2</math>.
|-
| 2004
| 0
| <math>y>0</math>
| <math>0 \leq x \leq 4.95 \times 10^{11}</math>
| [http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf Gourdon-Demichel]
| Numerical verification of RH & Riemann-von Mangoldt formula
| Results have not been independently verified
|-
| 2009
| <math>t > 0</math>
| <math>y > 0</math>
| <math>x \geq C(t)</math>
| [https://pure.tue.nl/ws/files/1769368/597490.pdf Ki-Kim-Lee]
| Theorem 1.3 of [https://pure.tue.nl/ws/files/1769368/597490.pdf Ki-Kim-Lee]
| <math>C(t)</math> is not given explicitly. Also they show <math>\Lambda < 1/2</math>.
|-
| 2017
| 0
| <math>y>0</math>
| <math>0 \leq x \leq 6.1 \times 10^{10}</math>
| [http://www.ams.org/journals/mcom/2017-86-307/S0025-5718-2017-03198-7/ 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>)
| [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493635 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>)
| [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
|
|-
| 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>)
| [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
| 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>)
| [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 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 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 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>)
| [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; argument principle
|
|-
| 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>; 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>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495062 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>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495065 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>
| [https://terrytao.wordpress.com/wp-admin/edit-comments.php 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
| 0.4
| <math>N \geq 3 \times 10^6 (x \geq 1.13 \times 10^{12})</math>
| [https://terrytao.wordpress.com/wp-admin/edit-comments.php KM]
| Analytic lower bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
|
|-
| Apr 7 2018
| 0.29
| <math>y \geq 0.29</math>
| <math>N \geq 19947</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495487 Anonymous]
| Triangle inequality bound on <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
| Would in principle show <math>\Lambda \leq 0.33205</math> if the matching barrier could be established
|-
| Apr 9 2018
| 0.2
| <math>y \geq 0.4</math>
| <math>N \geq 3 \times 10^5</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495723 Tao]
| Triangle inequality bound on <math>A^{eff}+B^{eff} / B^{eff}_0</math> and upper bounds on error terms
|
|-
| Apr 10 2018
| 0.2
| <math>y \geq 0.4</math>
| <math>4 \times 10^4 \leq N \leq 10^5; 100|N</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495841 KM]
| Euler2 mollifier and triangle inequality bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math>
| Error terms not estimated but look well within acceptable limits
|-
| Apr 12 2018
| 0.2
| <math>y \geq 0.4</math>
| <math>4 \times 10^4 \leq N \leq 3 \times 10^5</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495981 Anonymous]
| Euler2 mollifier and triangle inequality bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math>
| Error terms not estimated but look well within acceptable limits
|-
| Apr 12 2018
| 0.2
| <math>y \geq 0.4</math>
| <math>19947 \leq N \leq 4 \times 10^4</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-495981 Rudolph]
| Euler3 mollifier and triangle inequality bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math>
| Error terms not estimated but look well within acceptable limits
|-
| Apr 16 2018
| 0.2
| <math>y \geq 0.4</math>
| <math>19947 \leq N \leq 3 \times 10^5</math>
| [https://terrytao.wordpress.com/2018/03/28/polymath15-seventh-thread-going-below-0-48/#comment-496410 Rudolph]
| Estimating error terms in previous two ranges
| Completes proof of <math>\Lambda \leq 0.28</math>!
|-
| Apr 28 2018
| <math>0 \leq t \leq 0.2</math>
| <math>0.2 \leq y \leq 1</math>
| <math>6 \times 10^{10} + 2099 \leq x \leq 6 \times 10^{10} + 2100</math>
| [https://terrytao.wordpress.com/2018/04/17/polymath15-eighth-thread-going-below-0-28/#comment-497463 KM] / [https://terrytao.wordpress.com/2018/04/17/polymath15-eighth-thread-going-below-0-28/#comment-497481 Rudolph]
| Taylor series evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math>
|
|-
| May 1 2017
| <math>0 \leq t \leq 0.2</math>
| <math>0.2 \leq y \leq 1</math>
| <math>6 \times 10^{10} + 83952 - 1/2\leq x \leq 6 \times 10^{10} + 83952 + 1/2</math>
| [https://terrytao.wordpress.com/2018/04/17/polymath15-eighth-thread-going-below-0-28/#comment-497587 KM]
| Taylor series evaluation of <math>A^{eff}+B^{eff} / B^{eff}_0</math>
| Barrier chosen using Euler product heuristics to maximise lower bound
|-
| May 1 2017
| 0.2
| 0.2
| <math>69098 \leq N \leq 1.5 \times 10^6</math>
| [https://terrytao.wordpress.com/2018/04/17/polymath15-eighth-thread-going-below-0-28/#comment-497617 Rudolph]
| Euler5 mollifier and triangle inequality bounds on <math>A^{eff}+B^{eff} / B^{eff}_0</math>; only updating a few terms at a time when moving from <math>N</math> to <math>N+1</math>
| This should establish <math>\Lambda \leq 0.22</math>!
|}

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Polymath15 grant acknowledgments

Zero-free regions