k0 Table

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