Prime gaps in residue classes - where CSG > 1 is possible

[mart_r]

My monthly tribute to the world of number theory.
Largest CSG found during the past 4.3 weeks: 1.264846947 (p=113,109,089 / q=3,745,830)

                  max. p searched:
       q <= 1000: 1.908e+13
1000 < q <= 2690: 1.023e+13
2690 < q <= 1e+5: 1.770e+11
1e+5 < q <= 2e+5: 5.720e+10
2e+5 < q <= 5e+5: 4.600e+10
5e+5 < q <= 1e+6: 2.400e+10
1e+6 < q <= 2e+6: 1.20e+10
2e+6 < q <= 4e+6: 3.0e+9
 (only even q are examined)

P.S., @ Bobby: comparison between three ways of calculating CSG, in the case of q=3,613,418 / p₁=487,021:
g/[phi(q)*log²(p₂)]: 0.9241119774
my underappreciated formula: 1.0251848498
g/[phi(q)*log²(p₁)]: 2.2178622671
Finding a CSG above 2 by any other measure is, IMHO, impossible. But I have to be careful here since it's an open problem how large that value can actually be. Data might suggest that a global maximum depends on the ratio log(p)/log(q), in the sense that the largest CSG are attained when log(p)/log(q) is just a little above 1. It may be that CSG cannot be larger than, say, 1.2, if log(p)/log(q) is larger than 2 or thereabouts. All very sketchy at the moment, maybe I'll write a paper about it when the pandemic is over...

[Bobby Jacobs]

Originally Posted by mart_r

g/[phi(q)*log²(p₂)]: 0.9241119774
my underappreciated formula: 1.0251848498
g/[phi(q)*log²(p₁)]: 2.2178622671

I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

[mart_r]

Originally Posted by Bobby Jacobs

I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

Thanks for your support

[mart_r]

Search stats:

                  max. p searched:
       q <= 1000: 1.985e+13
1000 < q <= 2690: 1.211e+13
2690 < q <= 1e+5: 2.347e+11
1e+5 < q <= 2e+5: 6.245e+10
2e+5 < q <= 5e+5: 6.050e+10
5e+5 < q <= 1e+6: 2.400e+10
1e+6 < q <= 2e+6: 1.600e+10
2e+6 < q <= 5e+6: 3.000e+9
(only even q are examined)

The most recent inventory contained 2,849 exceptionally large gaps by the sum(Ri') measure,
of which 1,689 meet the conventional criterion g/[\(\varphi\)(q) log²(p+g)] > 1.

There is one new record by the conventional criterion:
p = 938,688,203
q = 4,200,826 = 2×7×61×4,919
k = 239
g/[φ(q) log²(p+g)] = 1.239732926499...
(unconventionally 1.2777045741...)

[mart_r]

Nothing to write home about anyway...

                  max. p searched:
       q <= 1000: 2.000e+13
1000 < q <= 2690: 1.489e+13
2690 < q <= 4566: 1.002e+12
4566 < q <= 1e+5: 2.755e+11
1e+5 < q <= 2e+5: 1.553e+11
2e+5 < q <= 5e+5: 8.300e+10
5e+5 < q <= 1e+6: 2.400e+10
1e+6 < q <= 2e+6: 2.100e+10
2e+6 < q <= 5e+6: 3.000e+9
5e+6 < q <= 1e+7: reserved
(only even q are examined)

[mart_r]

More data, dedicated to science.

While I'm at it, I also search for the largest instance per q of the least prime in an arithmetic progression p=k*q+r. The most recent work I could find on this is from Li, Pratt, and Shakan: https://arxiv.org/abs/1607.02543. They searched all q < 10⁶. I've searched even q < 4.4*10⁶ so far, so maybe this could be of use to somebody, anybody.
(There's > 40 MB of raw data, so if a special analysis is needed, feel free to ask.)

[mart_r]

Fooling around with the data for the first prime in an arithmetic progression, I noticed a peculiar pattern regarding the chance that p/[\(\varphi\)(q)*log²p] > x for increasing values of x.

Similar to the definitions in the Li-Pratt-Shakan paper, fix a positive integer q and let 0<r<q such that gcd(r,q)=1, let p_q,r denote the smallest prime number congruent to r mod q, and let p_q=max_0<r<q(p_q,r).

In the spirit of the Cramér-Shanks-Granville ratio, set CSG'_q=p_q/[\(\varphi\)(q)*log²p_q]. Li, Pratt, and Shakan work with the measure p_q/[\(\varphi\)(q)*log(\(\varphi\)(q))*log(q)], I'll just employ this as the LPS ratio, for comparison.

Simple question: how many times is CSG' or LPS larger than a certain value?

For the following table, three different clusters are considered:
Cluster A: even q<=4*10⁶ (2*10⁶ values)
Cluster B: even 2*10⁶<q<=4*10⁶ (10⁶ values)
Cluster C: even 3*10⁶<q<=4*10⁶ (5*10⁵ values)

CSG'> Clstr.A  Clstr.B  Clstr.C         LPS>  Clstr.A  Clstr.B  Clstr.C
0.40  1996627  1000000   500000         0.70  1999408  1000000   500000
0.41  1995088  1000000   500000         0.72  1999055  1000000   500000
0.42  1992609  1000000   500000         0.74  1998493   999999   499999
0.43  1988686  1000000   500000         0.76  1997413   999987   499994
0.44  1982698   999997   499999         0.78  1995016   999895   499964
0.45  1972980   999938   499988         0.80  1989661   999395   499771
0.46  1956792   999600   499911         0.82  1977830   997246   498898
0.47  1929962   997846   499402         0.84  1953466   990898   496135
0.48  1886001   991912   497385         0.86  1907725   975445   489161
0.49  1816387   975942   491430         0.88  1831562   944903   474813
0.50  1714427   943252   477919         0.90  1719430   894876   450743
0.51  1578487   889177   453937         0.92  1572026   824145   415979
0.52  1413135   812869   418167         0.94  1397497   735650   371858
0.53  1229672   719759   372964         0.96  1210737   637820   322624
0.54  1042771   618754   322976         0.98  1022744   538510   272569
0.55   864320   518285   272012         1.00   847021   444505   225041
0.56   702292   424195   223792         1.02   689066   359056   181708
0.57   561281   340760   180225         1.04   553090   286155   144757
0.58   443076   269818   143035         1.06   438807   225563   113833
0.59   346363   211428   112130         1.08   344928   175629    88314
0.60   268369   163734    86810         1.10   269623   135949    68199
0.61   206632   126025    66884         1.12   209657   104630    52301
0.62   158452    96452    51162         1.14   162771    80189    40029
0.63   120954    73441    39024         1.16   125565    61153    30475
0.64    91991    55748    29613         1.18    96885    46561    23153
0.65    69697    42174    22378         1.20    74325    35258    17445
0.66    52650    31751    16761         1.22    56975    26700    13090
0.67    39852    23966    12620         1.24    43949    20353    10037
0.68    30156    18103     9562         1.26    33727    15401     7584
0.69    22769    13603     7202         1.28    25905    11678     5710
0.70    17224    10247     5426         1.30    19890     8821     4333
0.71    13026     7662     4038         1.32    15250     6678     3261
0.72     9804     5776     3075         1.34    11735     5076     2493
0.73     7405     4359     2312         1.36     9063     3870     1898
0.74     5564     3247     1719         1.38     7002     2949     1439
0.75     4249     2489     1314         1.40     5420     2234     1086
0.76     3216     1881      979         1.42     4193     1693      813
0.77     2436     1396      718         1.44     3238     1273      593
0.78     1808     1027      526         1.46     2490      981      454
0.79     1360      777      403         1.48     1917      728      338
0.80     1020      583      302         1.50     1488      556      259
0.81      773      442      222         1.52     1148      419      193
0.82      578      326      156         1.54      876      316      137
0.83      437      241      118         1.56      670      232       99
0.84      311      164       80         1.58      509      169       74
0.85      237      125       62         1.60      397      123       56
0.86      181       89       45         1.62      313       91       42
0.87      140       71       34         1.64      249       73       32
0.88      100       53       27         1.66      191       55       26
0.89       78       39       21         1.68      148       40       19
0.90       58       28       16         1.70      118       28       13
0.91       46       21       14         1.72       89       24       13
0.92       32       15        8         1.74       71       17       11
0.93       28       14        8         1.76       57       12        8
0.94       24       12        8         1.78       43       11        7
0.95       14        7        6         1.80       32        6        5
0.96       12        5        4         1.82       28        5        4
0.97        8        2        1         1.84       24        4        3
0.98        6        2        1         1.86       20        2        1
0.99        4        1        0         1.88       16        1        0
1.00        2        1        0         1.90       15        1        0

For every increment of 0.01 in CSG', respectively 0.02 in LPS, the factor by which the values decrease is turned into the graphs in the attachment.
The heuristics of Li, Pratt, and Shakan suggest, or so I suppose, that eventually half of the time either LPS<1 or LPS>1.
Anyway I concentrate more on CSG' as this measure is more akin to the one used in dealing with the usual prime gaps. It's hard to compare LPS and CSG' directly (by trying to put them into relation or otherwise), also LPS differs more from my current analysis if odd q are taken into account than CSG', in the sense that CSG' is the same value for q and q/2 when q \(\equiv\) 2 (mod 4) (except in rare events when r=q/2+2), but LPS is not the same value for q and q/2.

Conjectures (provocative):
- (1) For every \(\varepsilon\)>0, the number of instances where CSG'<0.5-\(\varepsilon\) is finite.
- (2) The number of instances where CSG'<2/3 is asymptotic to q*whatchamercallit, where whatchamercallit (wc) is a constant in the vicinity of 1/35.
- (3) The number of instances where 2/3<=CSG'<=1 is asymptotic to q*wc*gubbins^(CSG'-2/3), where gubbins ~ 7*10^-13.
- (4) gubbins^(1/6) = wc/3 ??

Unknown:
- How does the graph continue past CSG'>1, provided we could calculate a large enough set of samples with larger q? A continuation of (3) for CSG'>1 (assuming it even holds for 2/3<=CSG'<=1) would imply arbitrarily large CSG values for prime gaps in AP, even if it's just for cases where log(p')/log(g) is small. OTOH, assuming a bounded CSG value, the factor by which the number of instances decreases per increment in CSG' has to decrease as well, further down the road.
- If (1) is true, then there's a point where asymptotically half of the time either CSG'>0.5+\(\delta\) or CSG'<0.5+\(\delta\). Whether or not \(\delta\) is a constant, who knows?

[mart_r]

                  max. p searched:
       q <= 1000: 2.000e+13
1000 < q <= 2690: 1.957e+13
2690 < q <= 4566: 5.851e+12
4566 < q <= 1e+5: 2.755e+11
1e+5 < q <= 2e+5: 1.764e+11
2e+5 < q <= 5e+5: 8.800e+10
5e+5 < q <= 1e+6: 2.400e+10
1e+6 < q <= 2e+6: 2.500e+10
2e+6 < q <= 6e+6: 4.000e+9
(only even q are examined;
 for q <= 4566, all first occurrence gaps are kept, for larger q only those with CSG > 1)

As for least primes in AP, there's an updated table in the attachment that includes odd q<2²¹. Though I believe the data to be correct, I don't give any guarantee that it be so.
A quantum computer would come in handy as many values of q and r could be tested in parallel. But there's nothing I can do...

[Bobby Jacobs]

Are CSG values greater than 2 possible?

[mart_r]

Originally Posted by Bobby Jacobs

Are CSG values greater than 2 possible?

Sorry for the delay, but I was waiting for new results from my workplace PC.
As far as I can see, we may not find a CSG much larger than 1.288... (see post # 28). I doubt that CSG > 1.4 is ever possible, but that's just a presumptuous guess.

Looking at the Plusle in your avatar, I'd like to talk about Pokémon instead...

[Bobby Jacobs]

I like Plusle because it has a plus sign, and + is used a lot in math. It is a math Pokemon. Plusle is also a bunny, and I like bunnies.