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Old 2021-08-17, 00:34   #19
ThomasK
 
Aug 2021

816 Posts
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In this post we will compare the Nullwertzahlen with the prime numbers. In particular, it is also about the comparison of the functions Pi (x) and W0(x).

Pi (x) is the number of prime numbers less than or equal to x and W0(x) is the number of Nullwertzahlen less than or equal to x.

To get a feel for why the Nullwertzahlen behave like prime numbers on the one hand, but also like inverse prime numbers on the other, here all prime numbers and all Nullwertzahlen up to 10000.


Prime Numbers up to 10000:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973.



Nullwertzahlen up to 10000:

2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 56, 60, 64, 72, 80, 84, 96, 108, 112, 120, 128, 132, 144, 160, 168, 176, 180, 192, 200, 208, 216, 224, 240, 252, 256, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 420, 432, 440, 448, 456, 468, 480, 504, 512, 520, 528, 540, 544, 552, 560, 576, 588, 600, 608, 612, 616, 624, 640, 648, 660, 672, 680, 684, 704, 720, 728, 736, 756, 760, 768, 780, 784, 792, 800, 816, 828, 832, 840, 864, 880, 896, 900, 912, 920, 924, 928, 936, 952, 960, 972, 992, 1000, 1008, 1020, 1024, 1040, 1044, 1056, 1064, 1080, 1088, 1092, 1104, 1116, 1120, 1140, 1152, 1160, 1176, 1188, 1200, 1216, 1224, 1232, 1240, 1248, 1260, 1280, 1288, 1296, 1320, 1344, 1360, 1368, 1380, 1392, 1400, 1404, 1408, 1428, 1440, 1452, 1456, 1472, 1480, 1488, 1500, 1512, 1520, 1536, 1560, 1568, 1584, 1596, 1600, 1620, 1624, 1632, 1656, 1664, 1680, 1716, 1728, 1736, 1740, 1760, 1764, 1776, 1792, 1800, 1824, 1836, 1840, 1848, 1856, 1860, 1872, 1904, 1920, 1932, 1936, 1944, 1960, 1968, 1980, 1984, 2000, 2016, 2040, 2048, 2052, 2064, 2072, 2080, 2088, 2100, 2112, 2128, 2160, 2176, 2184, 2200, 2208, 2220, 2232, 2240, 2244, 2256, 2268, 2280, 2288, 2296, 2304, 2320, 2340, 2352, 2368, 2376, 2400, 2408, 2432, 2436, 2448, 2460, 2464, 2480, 2484, 2496, 2508, 2520, 2560, 2576, 2580, 2592, 2600, 2604, 2624, 2632, 2640, 2664, 2688, 2700, 2704, 2720, 2736, 2744, 2752, 2760, 2772, 2784, 2800, 2808, 2816, 2820, 2856, 2880, 2904, 2912, 2916, 2940, 2944, 2952, 2960, 2968, 2976, 2992, 3000, 3008, 3024, 3036, 3040, 3060, 3072, 3080, 3096, 3108, 3120, 3132, 3136, 3168, 3180, 3192, 3200, 3240, 3248, 3264, 3276, 3280, 3300, 3312, 3328, 3344, 3348, 3360, 3384, 3392, 3400, 3420, 3432, 3440, 3444, 3456, 3472, 3480, 3520, 3528, 3536, 3540, 3552, 3564, 3584, 3600, 3612, 3640, 3648, 3672, 3680, 3696, 3712, 3720, 3744, 3760, 3776, 3780, 3800, 3808, 3816, 3828, 3840, 3864, 3872, 3888, 3900, 3904, 3920, 3936, 3948, 3952, 3960, 3968, 3996, 4000, 4032, 4048, 4056, 4080, 4092, 4096, 4104, 4116, 4128, 4140, 4144, 4160, 4176, 4200, 4212, 4224, 4240, 4248, 4256, 4284, 4312, 4320, 4352, 4356, 4368, 4392, 4400, 4416, 4428, 4440, 4452, 4464, 4480, 4488, 4500, 4512, 4536, 4560, 4576, 4592, 4600, 4608, 4620, 4640, 4644, 4680, 4704, 4720, 4736, 4752, 4760, 4784, 4788, 4800, 4816, 4824, 4840, 4860, 4864, 4872, 4880, 4884, 4896, 4920, 4928, 4956, 4960, 4968, 4992, 5000, 5016, 5040, 5076, 5088, 5096, 5100, 5104, 5112, 5120, 5124, 5148, 5152, 5160, 5184, 5200, 5208, 5220, 5248, 5264, 5280, 5292, 5304, 5320, 5328, 5360, 5376, 5400, 5408, 5412, 5440, 5456, 5460, 5472, 5488, 5504, 5508, 5520, 5544, 5568, 5580, 5600, 5616, 5628, 5632, 5640, 5664, 5676, 5680, 5700, 5712, 5720, 5724, 5760, 5796, 5800, 5808, 5824, 5832, 5840, 5856, 5880, 5888, 5904, 5920, 5928, 5936, 5940, 5952, 5964, 5984, 6000, 6016, 6032, 6048, 6072, 6080, 6084, 6120, 6132, 6144, 6156, 6160, 6192, 6200, 6204, 6216, 6240, 6264, 6272, 6300, 6320, 6336, 6360, 6372, 6384, 6400, 6432, 6440, 6448, 6468, 6480, 6496, 6512, 6528, 6552, 6560, 6588, 6600, 6608, 6624, 6636, 6656, 6660, 6664, 6688, 6696, 6720, 6732, 6760, 6768, 6776, 6784, 6800, 6804, 6816, 6832, 6840, 6864, 6880, 6888, 6900, 6912, 6936, 6944, 6960, 6972, 6996, 7000, 7008, 7020, 7040, 7056, 7072, 7080, 7104, 7128, 7140, 7168, 7176, 7200, 7216, 7224, 7236, 7260, 7280, 7296, 7308, 7320, 7344, 7360, 7380, 7392, 7400, 7424, 7440, 7448, 7452, 7480, 7488, 7500, 7504, 7520, 7524, 7552, 7560, 7568, 7584, 7600, 7616, 7632, 7644, 7656, 7668, 7680, 7696, 7728, 7740, 7744, 7752, 7776, 7788, 7800, 7808, 7812, 7840, 7872, 7884, 7896, 7904, 7920, 7936, 7952, 7956, 7968, 7980, 7992, 8000, 8008, 8040, 8052, 8064, 8096, 8100, 8112, 8120, 8160, 8176, 8184, 8192, 8200, 8208, 8232, 8256, 8272, 8280, 8288, 8316, 8320, 8352, 8360, 8400, 8424, 8448, 8460, 8480, 8496, 8512, 8520, 8528, 8532, 8544, 8568, 8576, 8580, 8600, 8624, 8640, 8664, 8680, 8700, 8704, 8712, 8736, 8748, 8760, 8784, 8800, 8820, 8832, 8840, 8844, 8848, 8856, 8880, 8892, 8904, 8928, 8944, 8960, 8964, 8976, 9000, 9016, 9024, 9048, 9072, 9088, 9108, 9120, 9152, 9180, 9184, 9200, 9216, 9240, 9248, 9280, 9288, 9296, 9300, 9324, 9328, 9344, 9360, 9372, 9384, 9396, 9400, 9408, 9440, 9464, 9472, 9480, 9504, 9520, 9540, 9568, 9576, 9600, 9612, 9632, 9636, 9648, 9660, 9672, 9680, 9720, 9728, 9744, 9760, 9768, 9776, 9792, 9800, 9828, 9840, 9856, 9880, 9900, 9912, 9920, 9936, 9960, 9968, 9984, 9996, 10000.



If you look at the distribution of the prime numbers as well as the distribution of the Nullwertzahlen, you will see amazing similarities on the one hand, but also some interesting differences on the other.


Number of divisors:

Prime number: 2
Nullwertzahl: at least C * ln n
___________________________________

Number of prim divisors:

Prime number: 1
Nullwertzahl: at least C * ln ln n
___________________________________

Construction:

Prim number: very difficult
Nullwertzahl: very easy
____________________________________

Determination of the prime divisors:

Prim number: very easy
Nullwertzahl: difficult
______________________________________

Congruence without the number 2:

Prim number: 1 mod 4 or 3 mod 4
Nullwertzahl: 0 mod 4
________________________________________

Difference of twins:

Prim number: 2
Nullwertzahl : 4
________________________________________

Construction of twins:

Prim number (p is prime und p + 2 is also prime): very difficult
Nullwertzahl (n is Nullwertzahl and n + 4 is also Nullwertzahl): very difficult
_________________________________________

Number of twins up to n:

Prim number: presumably O(n / (ln n)2)
Nullwertzahlen: presumably O(n / (ln n)2)
_____________________________________________

Distribution pattern locally:

Prim number: chaotic statistical
Nullwertzahlen: chaotic statistical plus Chaos surcharge for self-similarity reflection
____________________________________________

Distribution pattern global:

Prim number: regularly
Nullwertzahlen: regularly with greater fuzziness because of Chaos surcharge for self-similarity reflection
____________________________________________

Maximum distance between two immediate neighbors:

Prim number: presumably O((ln n)2)
Nullwertzahl: presumably O((ln n)2)


Note:

Even if the Riemann hypothesis, which has not yet been proven, should apply, it can be assumed that it can only be proven that the maximum distance between two immediately adjacent prime numbers in the number range up to n is at most O (n0.5), since the zeros of the continuation of the Riemannian Zeta function with real part 0.5 have a statistical background noise of C * n ^ 0.5 / ln n and also infinitely often for every constant C, however large, both R (x) - pi (x)> C * x [SUP] 0.5 [/ SUP] / ln x and pi (x) - R (x)> C * x [SUP] 0.5 [/ SUP] / ln x is forced. For the Nullwertzahlen plus the chaos surcharge for self-similarity reflection!


Direct comparison of the functions W0(n) and Pi (n) for the powers of ten from 1 to 12:


W0(10) = 3
Pi(10) = 4
W0(10) / Pi(10) = 0.75
_____________________________________________

W0(100) = 17
Pi(100) = 25
W0(100) / Pi(100) = 0.68
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W0(103) = 108
Pi(103) = 168
W0(103) / Pi(103) = 0.642857...
_____________________________________________

W0(104) = 755
Pi(104) = 1229
W0(104) / Pi(104) = 0.61432...
_____________________________________________

W0(105) = 5936
Pi(105) = 9592
W0(105) / Pi(105) = 0.618849...
_____________________________________________

W0(106) = 48474
Pi(106) = 78498
W0(106) / Pi(106) = 0.6175...
_____________________________________________

W0(107) = 406270
Pi(107) = 664579
W0(107) / Pi(107) = 0.611319...
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W0(108) = 3532031
Pi(108) = 5761455
W0(108) / Pi(108) = 0.61304...
_____________________________________________

W0(109) = 31295358
Pi(109) = 50847534
W0(109) / Pi(109) = 0.61547...
_____________________________________________

W0(1010) = 279591668
Pi(1010) = 455052511
W0(1010) / Pi(1010) = 0.6144...
_____________________________________________

W0(1011) = 2521429242
Pi(1011) = 4118054813
W0(1011) / Pi(108) = 0.612286...
_____________________________________________

W0(1012) = 22996137423
Pi(1012) = 37607912018
W0(1012) / Pi(1012) = 0.61147....
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