What are some nontrivial nonrandom properties of prime numbers. Consider the simple model where each number is prime with probability 1/log(n) by Montgomery and extensions of it. Once you add some simple modulo constraints it seems to capture very accurately many of the key properties, such as twin primes, linear sequences of primes, largest distance between primes and as far as I know most of the statistical properties of distances between primes. Are there any significant properties that it does not capture that are nontrivial and not easily explained by standard heuristics and we still don't understand.
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6$\begingroup$ Discussed in detail by Terence Tao here: terrytao.wordpress.com/2015/01/04/… Also Wikipedia: en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture $\endgroup$– Qiaochu YuanCommented Aug 12 at 4:14
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2$\begingroup$ @QiaochuYuan maybe even better to directly link to en.wikipedia.org/wiki/Maier%27s_theorem $\endgroup$– Sam HopkinsCommented Aug 12 at 17:40
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2$\begingroup$ A near-example is arxiv.org/abs/1603.03720 (but, as explained in that paper, the unusual regularity is ultimately explainable by "standard heuristics"). $\endgroup$– Terry TaoCommented Aug 12 at 23:33
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$\begingroup$ @TerryTao Thanks, that was one of the papers that motivated my question. $\endgroup$– ericfCommented Aug 13 at 15:51
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1$\begingroup$ One may have more examples if one looks at other areas of analytic number theory than the primes, such as the statistics of ranks of elliptic curves, where strong heuristic foundations are much more recent (see arxiv.org/abs/1602.01431, as well as the more recent work on murmurations) $\endgroup$– Terry TaoCommented Aug 13 at 18:16
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