Here is a table of k0 values for (pi_r*ϖ+delta_r*δ < 1) in different pi_r and delta_r settings, generated by a simple nonlinear optimizer.
Each link refers to a maple script ( based on the script in blog by Terence Tao) for a feasible k0. For each k0, the current narrowest H can be found in the database maintained by Andrew Sutherland.
- Here is the code K0Finder (Java, bash, and maple are required).
Parameters include varpi_v, deltap, and A, for ϖ = 1/pi_r-varpi_v*10-5, δ'=deltap, and A=A.
- Here are some additional notes for the usage (in mpz.txt):
(1) For setting the parameter range of varpi_v, varpi_v<105/pi_r should always be enforced.
(2) Sometimes there is an addditional constraint δ=δ', you just need to add a line "deltap := delta;", after the definition of delta (See this file).
(3) Sometimes (deltap - delta) instead of (deltap - delta)/2 is used for calculating thetat (See this file).
(4) the solution can be verified using maple, but do not run maple at the same directory when the algorithm is running.
- Computation for some other (pi_r, delta_r) pairs:
(828, 172), i=1: k0=20419.
(348, 68), i=1: k0=5447.
(168, 48), i=2: k0=1783 (The Deligne-avoiding case).
(148, 33), i=1: k0=1466.
(140, 32), i=1: k0=1346.
(116, 30), i=1: k0=1007.
(108, 30), i=1: k0=962.
(280/3, 80/3), i=1: k0=873 for δ=δ'; k0=720, if (deltap - delta) instead of (deltap - delta)/2 is used for calculating thetat; k0=720, if without any additional constraints.
(600/7, 180/7), i=4: k0=632 (need some modifications on the code).
- Note: the k0 values in the table below might not be optimal, but they might be good starting points for saving some computational time in searching better values in similar pi_r and delta_r settings.
pi_r, delta_r | 30 | 25 | 20 | 15 | 10 | 5 |
108 | 902 | 902 | 902 | 902 | 901 | 901 |
107 | 889 | - | - | - | - | 888 |
106 | 876 | - | - | - | - | 876 |
105 | 864 | - | - | - | - | 863 |
104 | 851 | - | - | - | - | 850 |
103 | 838 | - | - | - | - | 838 |
102 | 826 | - | - | - | - | 826 |
101 | 813 | - | - | - | - | 813 |
100 | 801 | 801 | 801 | 801 | 800 | 800 |
99 | 791 | - | - | - | - | 788 |
98 | 777 | - | - | - | - | 776 |
97 | 763 | - | - | - | - | 764 |
96 | 753 | - | - | - | - | 751 |
95 | 741 | - | - | - | - | 739 |
94 | 728 | - | - | - | - | 727 |
93 | 716 | - | - | - | - | 714 |
92 | 704 | - | - | - | - | 703 |
91 | 693 | - | - | - | - | 692 |
90 | 681 | - | - | - | - | 680 |
89 | 669 | - | - | - | - | 668 |
88 | 658 | - | - | - | - | 656 |
87 | 646 | - | - | - | - | 645 |
86 | 635 | - | - | - | - | 634 |
85 | 623 | - | - | - | - | 622 |
84 | 612 | - | - | - | - | 610 |
83 | 601 | - | - | - | - | 600 |
82 | 590 | - | - | - | - | 589 |
81 | 579 | - | - | - | - | 578 |
80 | 568 | - | - | - | - | 566 |
79 | 557 | - | - | - | - | 556 |
78 | 546 | - | - | - | - | 544 |
77 | 535 | - | - | - | - | 534 |
76 | 524 | - | - | - | - | 523 |
75 | 514 | 514 | 513 | 513 | 513 | 512 |
74 | 503 | - | - | - | - | 502 |
73 | 493 | - | - | - | - | 492 |
72 | 483 | - | - | - | - | 480 |
71 | 472 | - | - | - | - | 471 |
70 | 462 | - | - | - | - | 461 |
69 | 452 | - | - | - | - | 450 |
68 | 442 | - | - | - | - | 440 |
67 | 432 | - | - | - | - | 430 |
66 | 422 | - | - | - | - | 420 |
65 | 412 | - | - | - | - | 410 |
64 | 402 | - | - | - | - | 401 |
63 | 393 | - | - | - | - | 390 |
62 | 383 | - | - | - | - | 382 |
61 | 374 | - | - | - | - | 372 |
60 | 364 | - | - | - | - | 362 |
59 | 355 | - | - | - | - | 353 |
58 | 346 | - | - | - | - | 344 |
57 | 337 | - | - | - | - | 335 |
56 | 328 | - | - | - | - | 326 |
55 | 319 | - | - | - | - | 317 |
54 | 310 | - | - | - | - | 308 |
53 | 302 | - | - | - | - | 298 |
52 | 295 | - | - | - | - | 291 |
51 | 284 | - | - | - | - | 282 |
50 | 276 | 275 | 275 | 274 | 274 | 272 |
49 | 267 | - | - | - | - | 265 |
48 | 259 | - | - | - | - | 256 |
47 | 251 | - | - | - | - | 248 |
46 | 243 | - | - | - | - | 240 |
45 | 235 | - | - | - | - | 232 |
44 | 227 | - | - | - | - | 224 |
43 | 219 | - | - | - | - | 216 |
42 | 211 | - | - | - | - | 209 |
41 | 204 | - | - | - | - | 201 |
40 | 197 | - | - | - | - | 194 |
39 | 189 | - | - | - | - | 186 |
38 | 182 | - | - | - | - | 179 |
37 | 175 | - | - | - | - | 171 |
36 | 168 | - | - | - | - | 164 |
35 | 161 | - | - | - | - | 157 |
34 | 154 | - | - | - | - | 150 |
33 | 148 | - | - | - | - | 144 |
32 | 141 | - | - | - | - | 137 |
31 | 135 | - | - | - | - | 131 |
30 | 129 | - | - | - | - | 124 |
29 | 123 | - | - | - | - | 118 |
28 | 117 | - | - | - | - | 112 |
27 | 111 | - | - | - | - | 106 |
26 | 105 | - | - | - | - | 100 |
25 | 100 | 99 | - | 96 | - | 94 |