A 15 year old or so F Gauss saw a pattern in the distribution of primes.......while a "good ball park figure"..........if that wasn't a code.......I don't know what would be......but anyways...............it was overall a decent average......but locally irregular..............while the denominator goes up by one..............the fraction goes down.............b/c it is a inverse relationship.............1/4 is smaller numerically than 1/2..............1/2 = 0.5 and 1/4 = 0.25....................50 cents is more than 25 cents.................................and while the denominator goes up (the number on the bottom).........uniformly one fraction to the next..........by ones...............the values go down.................erratically.............

Table of π(x), x / log x, and li(x)[edit]

The table compares exact values of π(x) to the two approximations x / log x and li(x). The last column, x / π(x), is the average prime gap below x.

x	π(x)	π(x) − x/log x	π(x)/x / log x	li(x) − π(x)	x/π(x)
10	4	−0.3	0.921	2.2	2.500
102	25	3.3	1.151	5.1	4.000
103	168	23.0	1.161	10.0	5.952
104	7003122900000000000♠1229	143.0	1.132	17.0	8.137
105	7003959200000000000♠9592	906.0	1.104	38.0	10.425
106	7004784980000000000♠78498	7003611600000000000♠6116.0	1.084	130.0	12.740
107	7005664579000000000♠664579	7004441580000000000♠44158.0	1.071	339.0	15.047
108	7006576145500000000♠5761455	7005332774000000000♠332774.0	1.061	754.0	17.357
109	7007508475340000000♠50847534	7006259259200000000♠2592592.0	1.054	7003170100000000000♠1701.0	19.667
1010	7008455052511000000♠455052511	7007207580290000000♠20758029.0	1.048	7003310400000000000♠3104.0	21.975
1011	7009411805481300000♠4118054813	7008169923159000000♠169923159.0	1.043	7004115880000000000♠11588.0	24.283
1012	7010376079120180000♠37607912018	7009141670519300000♠1416705193.0	1.039	7004382630000000000♠38263.0	26.590
1013	7011346065536839000♠346065536839	7010119928584520000♠11992858452.0	1.034	7005108971000000000♠108971.0	28.896
1014	7012320494175080200♠3204941750802	7011102838308636000♠102838308636.0	1.033	7005314890000000000♠314890.0	31.202
1015	7013298445704226690♠29844570422669	7011891604962452000♠891604962452.0	1.031	7006105261900000000♠1052619.0	33.507
1016	7014279238341033925♠279238341033925	7012780428984439300♠7804289844393.0	1.029	7006321463200000000♠3214632.0	35.812
1017	7015262355715765423♠2623557157654233	7013688837346932810♠68883734693281.0	1.027	7006795658900000000♠7956589.0	38.116
1018	7016247399542877408♠24739954287740860	7014612483070893536♠612483070893536.0	1.025	7007219495550000000♠21949555.0	40.420
1019	7017234057667276344♠234057667276344607	7015548162416936996♠5481624169369960.0	1.024	7007998777750000000♠99877775.0	42.725
1020	7018222081960256091♠2220819602560918840	7016493471930446597♠49347193044659701.0	1.023	7008222744644000000♠222744644.0	45.028
1021	7019211272694860187♠21127269486018731928	7017446579871578168♠446579871578168707.0	1.022	7008597394254000000♠597394254.0	47.332
1022	7020201467286689315♠201467286689315906290	7018406070400601962♠4060704006019620994.0	1.021	7009193235520800000♠1932355208.0	49.636
1023	7021192532039160680♠1925320391606803968923	7019370835137665786♠37083513766578631309.0	1.020	7009725018621600000♠7250186216.0	51.939
1024	7022184355997673492♠18435599767349200867866	7020339996354713708♠339996354713708049069.0	1.019	7010171469072780000♠17146907278.0	54.243
1025	7023176846309399143♠176846309399143769411680	7021312851663784303♠3128516637843038351228.0	1.018	7010551609809390000♠55160980939.0	56.546
OEIS	A006880	A057835		A057752

The value for π(10²⁴) was originally computed assuming the Riemann hypothesis;^[28] it has since been verified unconditionally.^[29]

Analogue for irreducible polynomials over a finite field[edit]

There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem.
To state it precisely, let F = GF(q) be the finite field with q elements, for some fixed q, and