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Zhang's celebrated result established the existence of bounded gaps between primes, that is, there exists a constant $B$ (in Zhang's paper $B$ can be taken to be $7 \times 10^7$, and this was almost immediately improved significantly so that one may take $B= 4680$) such that there are infinitely many primes $p_n$ such that $p_n - p_{n-1} < B$.

My question considers the opposite case. Do there exist, infinitely often, gaps that are much larger than average? The prime number theorem implies that the average gap between $p_n$ and $p_{n+1}$ is about $\log p_n$.

Thus, for which constant $C > 1$ can one establish the existence of infinitely many primes $p_n$ such that $p_n - p_{n-1} > C\log p_n$? And what is the best known constant to date?

Edit: in view of the wikipedia entry on this topic, it seems that the correct thing to look at is

$$\displaystyle p_n - p_{n-1} > \frac{c \log n \log \log n \log \log \log \log n}{(\log \log \log n)^2}.$$ This seems like a rather unnatural function, but for now it is not known if the constant $c$ in the above inequality may be taken to be arbitrarily large. Is this inequality expected to be the right order of magnitude?

Thanks for any insight.

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    $\begingroup$ The Wikipedia article en.wikipedia.org/wiki/Prime_gap on prime gaps covers this topic. In particular, C can be taken to be arbitrarily large. $\endgroup$
    – rlo
    Nov 18, 2013 at 19:35
  • $\begingroup$ Judging by Terry Tao's most recent post on his blog, it seems that a mathematician named Maynard could show that one may take $B=600$. $\endgroup$
    – Sylvain JULIEN
    Nov 18, 2013 at 19:39
  • $\begingroup$ My personal reading list in this area includes Jacobsthal's function. In addition to Westzynthius, Rankin, Erdos, Pomerance, Maier, Pintz, and a host of others working on prime gaps, I recommend Jacobsthal, Vaughan, Stevens, Iwaniec, and Hagedorn in reading about large gaps between totatives. The estimate above seems hard to beat, but even showing c(log n)^2 is an upper bound is hard. I hope to submit some related ArXiv articles soon. Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2013.11.18 $\endgroup$
    – Gerhard Paseman
    Nov 18, 2013 at 19:52
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    $\begingroup$ Following up on @GerhardPaseman's comment, the expected order of magnitude is $c \log^2 n$ (though proving this is hard!). See, for example, this paper of Granville: www.dms.umontreal.ca/~andrew/PDF/cramer.pdf. It discusses Cramer's conjecture, why it's probably wrong (at least in a precise form), and what the correct modification should be. $\endgroup$
    – rlo
    Nov 18, 2013 at 19:55

2 Answers 2

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I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639.$ I believe Cramer-Granville is the conjecture that $ \; \limsup g/\log^2 p$ is nonzero but finite, and the disagreement is over whether it is more likely to be $1$ or something else. However, it gives an opinion on your original question. 

   Stolen from 
   http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm

   the size of the gap is g

   next are the number of decimal digits in p

   for 4 * 10^18 > p >= 11, g < log^2 p = (log p)^2.

   Oh, logarithms base e == 2.718281828459


   ==================================
            g   digits of p           p    log p   g/log p  g/log^2 p
     1      1   1                      2 0.693147   1.4427    2.08137
     2      2   1                      3  1.09861  1.82048    1.65707
     3      4   1                      7  1.94591  2.05559    1.05637
     4      6   2                     23  3.13549  1.91357   0.610294
     5      8   2                     89  4.48864  1.78228   0.397065
     6     14   3                    113  4.72739  2.96147   0.626449
     7     18   3                    523  6.25958  2.87559    0.45939
     8     20   3                    887  6.78784  2.94644   0.434076
     9     22   4                   1129  7.02909  3.12985   0.445271
    10     34   4                   1327  7.19068  4.72835   0.657566
    11     36   4                   9551   9.1644  3.92824   0.428642
    12     44   5                  15683  9.66033  4.55471   0.471486
    13     52   5                  19609  9.88374  5.26116   0.532305
    14     72   5                  31397  10.3545  6.95352   0.671548
    15     86   6                 155921  11.9571  7.19238   0.601515
    16     96   6                 360653  12.7957  7.50254   0.586334
    17    112   6                 370261   12.822  8.73501   0.681254
    18    114   6                 492113  13.1065    8.698   0.663642
    19    118   7                1349533  14.1153  8.35974   0.592248
    20    132   7                1357201  14.1209  9.34782   0.661983
    21    148   7                2010733   14.514   10.197   0.702566
    22    154   7                4652353  15.3529  10.0307   0.653342
    23    180   8               17051707  16.6518  10.8097   0.649161
    24    210   8               20831323   16.852  12.4615   0.739466
    25    220   8               47326693  17.6726  12.4487   0.704405
    26    222   9              122164747  18.6209  11.9221   0.640254
    27    234   9              189695659  19.0609  12.2764   0.644062
    28    248   9              191912783  19.0726   13.003   0.681764
    29    250   9              387096133  19.7742  12.6427   0.639356
    30    282   9              436273009  19.8938  14.1753   0.712549
    31    288  10             1294268491  20.9812  13.7266   0.654231
    32    292  10             1453168141   21.097  13.8408   0.656056
    33    320  10             2300942549  21.5566  14.8447   0.688637
    34    336  10             3842610773  22.0694  15.2247   0.689855
    35    354  10             4302407359  22.1824  15.9586   0.719423
    36    382  11            10726904659   23.096  16.5396   0.716125
    37    384  11            20678048297  23.7523  16.1668   0.680642
    38    394  11            22367084959  23.8309  16.5332   0.693772
    39    456  11            25056082087  23.9444  19.0441   0.795349
    40    464  11            42652618343  24.4764  18.9571   0.774506
    41    468  12           127976334671  25.5751   18.299   0.715502
    42    474  12           182226896239  25.9285   18.281   0.705055
    43    486  12           241160624143  26.2087  18.5434   0.707529
    44    490  12           297501075799  26.4187  18.5475   0.702059
    45    500  12           303371455241  26.4382   18.912   0.715328
    46    514  12           304599508537  26.4423  19.4386   0.735133
    47    516  12           416608695821  26.7554  19.2858   0.720819
    48    532  12           461690510011  26.8582  19.8078   0.737495
    49    534  12           614487453523  27.1441  19.6728   0.724756
    50    540  12           738832927927  27.3283  19.7597   0.723048
    51    582  13          1346294310749  27.9284   20.839   0.746159
    52    588  13          1408695493609  27.9737  21.0198   0.751412
    53    602  13          1968188556461  28.3081   21.266   0.751232
    54    652  13          2614941710599  28.5923  22.8034   0.797536
    55    674  13          7177162611713  29.6019  22.7688   0.769166
    56    716  14         13829048559701  30.2578  23.6633   0.782057
    57    766  14         19581334192423  30.6056  25.0281   0.817762
    58    778  14         42842283925351  31.3885  24.7861   0.789655
    59    804  14         90874329411493  32.1405  25.0152   0.778307
    60    806  15        171231342420521   32.774  24.5926   0.750369
    61    906  15        218209405436543  33.0165  27.4408   0.831126
    62    916  16       1189459969825483  34.7123  26.3884   0.760203
    63    924  16       1686994940955803  35.0617  26.3535   0.751632
    64   1132  16       1693182318746371  35.0654  32.2825   0.920639
    65   1184  17      43841547845541059  38.3194  30.8982   0.806335
    66   1198  17      55350776431903243  38.5525  31.0745   0.806032
    67   1220  17      80873624627234849  38.9317   31.337   0.804922
    68   1224  18     203986478517455989  39.8568  30.7099   0.770506
    69   1248  18     218034721194214273  39.9234  31.2598   0.782995
    70   1272  18     305405826521087869  40.2604  31.5943   0.784749
    71   1328  18     352521223451364323  40.4039  32.8681   0.813489
    72   1356  18     401429925999153707  40.5338  33.4536   0.825325
    73   1370  18     418032645936712127  40.5743  33.7652   0.832181
    74   1442  18     804212830686677669  41.2286  34.9757   0.848335
    75   1476  19    1425172824437699411  41.8008  35.3103   0.844728
          g   digits of p             p    log p   g/log p  g/log^2 p
   ==================================
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Erdos's problem, which asks whether or not the bound

$$\displaystyle p_n - p_{n-1} > \frac{c \log n \log \log n \log \log \log \log n}{(\log \log \log n)^2}$$

holds with $c$ arbitrarily large, has been answered in the positive independently by Ben Green, Kevin Ford, Sergei Konyagin, and Terry Tao and James Maynard. However, it seems that they have decided to write a single paper on the subject, which is found here: http://arxiv.org/abs/1412.5029

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    $\begingroup$ You might note that they removed a factor of logloglogn from the denominator. That may inspire more click throughs to the abstract. Gerhard "Think Of It As Teaser" Paseman, 2015.05.14 $\endgroup$
    – Gerhard Paseman
    May 15, 2015 at 4:17
  • $\begingroup$ When you say "the bound (...) holds for $c$ arbitrary large", do you mean "for every $c$ there are infinitely any $n$ satisfying the bound (...)" or something else? $\endgroup$
    – YCor
    May 15, 2015 at 9:03
  • $\begingroup$ This implies that the left hand can be compared to the right hand with a number chosen arbitrarily it will always be exceeded with growing $n$, a natural number indicating the difference between prime. So for a usual math person's intuition is $c==1$ for such comparisons, but for math geeks this is a special. They prefer not to divide the nice difference with the term in $n$ for reasons of beauty. If you do this in through this, gets meaningful and understandable in terms of a diverging term on the left. Then put left a limit sign and right the infinity symbol, and this is Analysis I. $\endgroup$
    – Steffen Jaeschke
    Feb 28, 2023 at 19:44

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