On the lower integration limit

The conjectured frequency of occurrence of a prime gap of size 2*r is given by the integral :

Eq.(1)   

Here the lower integration limit is traditionally set to a=2. For low values of "r" this value is maybe acceptable but an unwelcome features which we show on the example of r=10 can be discerned.

Figure 1: Conjecture gap distribution, r=10 , a=2
Fig. 1 shows a typical gap distribution, here half-gap size r = 10, except for the rapid increase near x=2. A closer look , see Fig. 2, shows that this initial increase is confined to very small values of x followed by a plateau.
Figure 2: Conjecture gap distribution, r=10 , a=2

Actually, as an even closer look reveals as in Fig. 3, there is a slight undulation on the curve with minima and maxima according to those real roots of the integrand in the range x > 2.

Figure 3: Conjecture gap distribution, r=10 , a=2

For r=10 we have N=6 because T_{10,k}=0 for k>6 and the integrand of Eq.(1) can be re-written with the substitution u=1/ln(x) as :

A(10,1) + A(10,2)*u + A(10,3)*u^2 + A(10,4)*u^3 + A(10,5)*u^4 + A(10,6)*u^5

save for a factor of u^2. This polynomial has 5 real roots at x = 1.47 , 6.54 , 25.85 , 93.46 , and 323.38, respectively.

This behavior, a rapid change at small value of x followed by a "plateau" is observed at all values of the half-gap size r with the extent and level of the plateau changing with r as shown in Table 1. The extent of the plateau as given in the columns xMin and xMax was determined by evaluating Eq.(1) and detecting the smallest value of x (=xMin) for which the outcome of the integral in Eq.(1) does not differ by more than 1 in magnitude from the value at x=2*r which itself is listed in column "Level". xMax is found in analog fashion. In column "Trail" we list the trailing prime of the actual first occurrence of each prime gap.

At large values of "r" the level of the plateau becomes certainly unacceptable because of its magnitude and sign. This feature can be removed by choosing the lower integration limit in Eq.(1) different from 2 and for given r any value within the extent of its plateau without any consequences for the conjectured prime gap count.

Our present (June 2011) choice is :

a = 2*r

taking into consideration the trend of xMin to increase as r increases.

r LevelxMinxMaxTrail
3-7.146811e+00 3.75 25.84 29
4-6.780711e-01 3.38 166.92 97
53.555250e+00 2.30 160.33 149
67.269052e+00 4.70 319.69 211
7-1.500969e+01 2.67 493.72 127
8-1.740245e+01 3.29 816.22 1847
9-1.269525e+01 5.10 772.43 541
101.191173e+02 4.03 1744.61 907
112.158129e+02 4.48 1640.82 1151
12-7.248731e+02 3.84 1893.96 1693
131.717290e+02 2.56 3780.65 2503
14-5.686264e+01 3.99 3759.47 2999
151.000689e+03 5.71 3978.24 4327
161.400761e+02 4.12 9369.24 5623
17-1.462663e+02 4.69 9180.64 1361
18-1.299790e+03 5.14 8622.22 9587
19-1.025210e+03 4.68 15426.91 30631
206.891315e+03 4.95 14741.76 19373
212.125382e+04 4.53 15070.38 16183
22-2.513967e+04 4.87 27626.70 15727
23-8.400910e+04 5.41 32971.55 81509
243.393150e+04 5.44 27167.92 28277
255.473856e+04 5.18 41592.90 31957
263.435598e+05 5.80 54752.77 19661
27-5.407648e+05 4.77 43055.14 35671
283.276175e+05 5.41 70972.27 82129
29-1.056187e+05 5.29 77617.41 44351
30-1.591548e+06 6.59 63905.57 43391
31-1.584449e+06 5.53 139456.23 34123
326.731216e+06 5.48 132553.30 89753
33-5.694673e+06 6.46 109335.37 162209
341.315161e+06 5.64 197733.80 134581
35-4.491112e+06 5.81 168493.77 173429
366.497845e+07 6.48 192120.13 31469
371.306555e+07 5.68 286548.68 404671
38-1.918844e+07 6.18 333046.34 212777
395.630535e+07 6.26 246452.28 188107
40-6.515843e+08 6.56 393129.71 542683
41-4.072055e+08 5.79 497162.49 265703
424.508348e+09 6.85 355235.78 461801
434.187576e+09 6.72 682685.20 156007
441.692225e+09 6.00 684344.25 544367
45-4.863103e+10 6.85 506837.49 404941
46-1.542617e+10 6.60 978211.87 927961
476.217293e+10 6.16 1053916.41 1101071
483.942936e+10 5.68 818269.93 360749
491.557290e+11 6.76 1236905.77 604171
50-4.066972e+10 6.86 1248928.23 396833
51-6.502928e+10 6.86 1160471.66 1444411
52-2.992801e+11 6.08 1826045.43 1388587
53-1.429462e+10 6.41 2042790.95 1098953
542.161930e+12 7.49 1579940.86 2238931
55-4.167739e+12 7.38 2228797.64 1468387
563.037950e+12 7.14 2594428.37 370373
57-1.227498e+13 7.77 2162702.30 492227
581.549005e+13 7.00 3601820.05 5845309
594.656799e+12 7.34 3663246.00 1349651
60-8.686581e+12 6.77 2627948.74 1895479
614.004533e+13 6.53 5158412.00 3117421
62-3.697010e+13 7.24 5126178.80 6752747
631.140574e+14 7.53 3844819.58 1671907
64-1.955935e+13 7.45 7093724.88 3851587
65-1.280174e+15 7.77 6067252.22 5518817
66-2.368601e+14 7.84 5577981.27 1357333
67-1.110395e+15 7.53 9118478.38 6958801
689.541472e+15 7.26 9902638.84 6371537
696.888115e+15 7.33 7228777.76 3826157
70-1.942394e+16 7.98 9831332.50 7621399
71-2.952550e+16 7.37 13243835.22 10343903
725.892939e+16 8.22 10180490.13 11981587
738.871520e+16 8.01 16854125.74 6034393
74-6.800904e+15 7.91 15709806.29 2010881
75-3.018136e+17 8.23 11327265.88 13626407
76-2.877170e+17 7.87 21934747.99 8421403
774.433099e+17 8.23 19566417.76 4652507
787.309582e+17 8.07 16967945.89 17983873
791.092235e+17 8.39 28031385.44 49269739
80-1.738574e+18 8.48 25939568.79 33803849
813.880588e+18 8.41 23178503.23 39175379
82-4.718240e+18 7.86 34824673.71 20285263
83-1.257776e+19 7.63 38610198.32 83751287
842.552417e+19 8.79 26187071.47 37305881
86-3.516789e+19 8.34 51572181.16 38394299
873.892571e+20 8.69 37247538.61 52721287
89-2.304191e+20 8.37 60871128.67 39390167
90-3.794884e+20 8.45 41846106.28 17051887
919.884978e+20 7.82 72183561.99 36271783
927.331994e+20 8.73 78991463.84 79167917
93-5.781004e+21 8.18 62087650.87147684323
94-9.335560e+21 8.27 103216687.91134066017
97-1.114545e+22 8.78 125830655.68166726561
100-9.518484e+22 8.64 139192231.38378044179
Table 1 : Characteristics of "plateau" as function of r
and actual first occurrence of prime gap 2*r


Zig Herzog; hgn@psu.edu
Last revised: 06/26/11