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Information about Why we believe that riesel is wrong about prime probabilities

In this text we will explain our opinion why we believe that Hans Riesel is wrong about prime probabilities in between formula 3.8 and 3.9 of his book "Prime Numbers and Computer Methods for Factorization"

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I don't believe in more or less efficient sieves for the purpose of prime counting. You either have a sieve that works or one that doesn't. Besides: my reasoning classified by some as a “random sieve” is in fact nothing more than a reverse usage of the sieve of Eratosthenes, inclusive my “one and only one number, zero” which is struck by all the sieving primes. This number zero is as a matter of fact at the origin of all my disliked sine waves ... As a bonus, I would like to elaborate briefly on possible subtleties happening between sqrt(x) and x. Therefore, I am making a little case study where I study the primes 2, 3, 5 and 7 and we will investigate what happens between 7 and 49. We take the first prime 2 and imagine what influence he could have on prime probabilities on a detailed level. On every multiple of 2, the probability to find a new prime will clearly be 0%. However inbetween these multiples, this prime should not really bother the formation of new primes. Suppose that we can then represent this probability as abs(sin(pi()*x/2)). We take a similar reasoning for the probabilities caused by the primes 3, 5 and 7, sum up all these probabilities and divide by the number of primes and come to a function: probability(x)=(abs(sin(pi()*x/2))+abs(sin(pi()*x/3))+abs(sin(pi()*x/5))+abs(sin(pi()*x/7)))/4 This function can be seen in following graph:

What is interesting in this graph is, that there seem to be maxima in the neighborhood of: 11, 13, 17, 23, 31, 37, 43 and 47 which are all primes!. So our reasoning is probably not that stupid. I added a trendline to this graph and what we can notice is that it only has a very small negative slope, meaning that the probability to find primes only slowly decreases between sqrt(x) and x. Contrary to what Riesel thought and to what myself also had expected somewhat: nothing really happens between sqrt(x) and x in absolute values of the average probabilities, except some predictions on the positions of new primes. Appendix: 2 cases studies indicating that my formula's are correct.

primes ind_prob tot_prob range prime chances check Vacancy count: average: 2 3 0,5 0,666667 0,229 100 22,9 1 2 22 21 23 5 7 0,8 0,85714 3 25 4 24 5 23 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 count: 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 2th range of 100 49 50 59 60 69 70 79 80 89 90 99 100 109 110 119 120 129 130 139 140 51 61 71 81 91 101 111 121 131 141 52 62 72 82 92 102 112 122 132 142 53 63 73 83 93 103 113 123 133 143 54 64 74 84 94 104 114 124 134 144 55 65 75 85 95 105 115 125 135 145 56 66 76 86 96 106 116 126 136 146 57 67 77 87 97 107 117 127 137 147 58 68 78 88 98 108 118 128 138 148 count: 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 21 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 3rd range of 100 100 101 110 111 120 121 130 131 140 141 150 151 160 161 170 171 180 181 190 191 102 112 122 132 142 152 162 172 182 192 103 113 123 133 143 153 163 173 183 193 104 114 124 134 144 154 164 174 184 194 105 115 125 135 145 155 165 175 185 195 106 116 126 136 146 156 166 176 186 196 107 117 127 137 147 157 167 177 187 197 108 118 128 138 148 158 168 178 188 198 109 119 129 139 149 159 169 179 189 199 count: 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 25 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 4th range of 100 1001 1002 1011 1012 1021 1022 1031 1032 1041 1042 1051 1052 1061 1062 1071 1072 1081 1082 1091 1092 1003 1013 1023 1033 1043 1053 1063 1073 1083 1093 1004 1014 1024 1034 1044 1054 1064 1074 1084 1094 1005 1015 1025 1035 1045 1055 1065 1075 1085 1095 1006 1016 1026 1036 1046 1056 1066 1076 1086 1096 1007 1017 1027 1037 1047 1057 1067 1077 1087 1097 1008 1018 1028 1038 1048 1058 1068 1078 1088 1098 1009 1019 1029 1039 1049 1059 1069 1079 1089 1099 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 count: 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 5th range of 100 10017 10018 10027 10028 10037 10038 10047 10048 10057 10058 10067 10068 10077 10078 10087 10088 10097 10098 10107 10108 10019 10029 10039 10049 10059 10069 10079 10089 10099 10109 10020 10030 10040 10050 10060 10070 10080 10090 10100 10110 10021 10031 10041 10051 10061 10071 10081 10091 10101 10111 10022 10032 10042 10052 10062 10072 10082 10092 10102 10112 10023 10033 10043 10053 10063 10073 10083 10093 10103 10113 10024 10034 10044 10054 10064 10074 10084 10094 10104 10114 10025 10035 10045 10055 10065 10075 10085 10095 10105 10115 10026 10036 10046 10056 10066 10076 10086 10096 10106 10116 count: 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 23 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1st range of 100 1 11 21 31 41 51 61 71 81 91

primes ind_prob tot_prob range prime chances check Vacancy count: average: 3 0,67 0,505 100 50,5 1 50 50 7 13 0,86 0,9231 23 0,96 2 50 3 52 4 50 5 48 1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100 2th range of 100 49 59 69 79 89 99 109 119 129 139 50 60 70 80 90 100 110 120 130 140 51 61 71 81 91 101 111 121 131 141 52 62 72 82 92 102 112 122 132 142 53 63 73 83 93 103 113 123 133 143 54 64 74 84 94 104 114 124 134 144 55 65 75 85 95 105 115 125 135 145 56 66 76 86 96 106 116 126 136 146 57 67 77 87 97 107 117 127 137 147 58 68 78 88 98 108 118 128 138 148 3rd range of 100 100 110 120 130 140 150 160 170 180 190 101 111 121 131 141 151 161 171 181 191 102 112 122 132 142 152 162 172 182 192 103 113 123 133 143 153 163 173 183 193 104 114 124 134 144 154 164 174 184 194 105 115 125 135 145 155 165 175 185 195 106 116 126 136 146 156 166 176 186 196 107 117 127 137 147 157 167 177 187 197 108 118 128 138 148 158 168 178 188 198 109 119 129 139 149 159 169 179 189 199 4th range of 100 1001 1011 1021 1031 1041 1051 1061 1071 1081 1091 1002 1012 1022 1032 1042 1052 1062 1072 1082 1092 1003 1013 1023 1033 1043 1053 1063 1073 1083 1093 1004 1014 1024 1034 1044 1054 1064 1074 1084 1094 1005 1015 1025 1035 1045 1055 1065 1075 1085 1095 1006 1016 1026 1036 1046 1056 1066 1076 1086 1096 1007 1017 1027 1037 1047 1057 1067 1077 1087 1097 1008 1018 1028 1038 1048 1058 1068 1078 1088 1098 1009 1019 1029 1039 1049 1059 1069 1079 1089 1099 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 5th range of 100 10017 10027 10037 10047 10057 10067 10077 10087 10097 10107 10018 10028 10038 10048 10058 10068 10078 10088 10098 10108 10019 10029 10039 10049 10059 10069 10079 10089 10099 10109 10020 10030 10040 10050 10060 10070 10080 10090 10100 10110 10021 10031 10041 10051 10061 10071 10081 10091 10101 10111 10022 10032 10042 10052 10062 10072 10082 10092 10102 10112 10023 10033 10043 10053 10063 10073 10083 10093 10103 10113 10024 10034 10044 10054 10064 10074 10084 10094 10104 10114 10025 10035 10045 10055 10065 10075 10085 10095 10105 10115 10026 10036 10046 10056 10066 10076 10086 10096 10106 10116 1st range of 100 count: 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 50 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 50 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 52 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 50 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1 48 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1 0 count: count: count: count:

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