By way of analogy with the prime counting function , the notation denotes the number of primes of the form less than or equal to (Shanks 1993, pp. 21-22).
Hardy and Littlewood proved that an switches leads infinitely often, a result known as the prime quadratic effect. The bias of the sign of is known as the Chebyshev bias.
Groups of equinumerous values of include (, ), (, ), (, , , ), (, ), (, , , , , ), (, , , ), (, , , , , ), and so on. The values of for small are given in the following table for the first few powers of ten (Shanks 1993).
Sloane | A091115 | A091116 | A091098 | A091099 |
1 | 2 | 1 | 2 | |
11 | 13 | 11 | 13 | |
80 | 87 | 80 | 87 | |
611 | 617 | 609 | 619 | |
4784 | 4807 | 4783 | 4808 | |
39231 | 39266 | 39175 | 39322 | |
332194 | 332384 | 332180 | 332398 | |
2880517 | 2880937 | 2880504 | 2880950 | |
25422713 | 25424820 | 25423491 | 25424042 |
Sloane | A091120 | A091121 | A091122 | A091123 | A091124 | A091125 |
0 | 1 | 1 | 0 | 1 | 0 | |
3 | 4 | 5 | 3 | 5 | 4 | |
28 | 27 | 30 | 26 | 29 | 27 | |
203 | 203 | 209 | 202 | 211 | 200 | |
1593 | 1584 | 1613 | 1601 | 1604 | 1596 | |
13063 | 13065 | 13105 | 13069 | 13105 | 13090 | |
110653 | 110771 | 110815 | 110776 | 110787 | 110776 | |
960023 | 960114 | 960213 | 960085 | 960379 | 960640 | |
8474221 | 8474796 | 8475123 | 8474021 | 8474630 | 8474742 |
Sloane | A091126 | A091127 | A091128 | A091129 |
0 | 1 | 1 | 1 | |
5 | 7 | 6 | 6 | |
37 | 44 | 43 | 43 | |
295 | 311 | 314 | 308 | |
2384 | 2409 | 2399 | 2399 | |
19552 | 19653 | 19623 | 19669 | |
165976 | 166161 | 166204 | 166237 | |
1439970 | 1440544 | 1440534 | 1440406 | |
12711220 | 12712340 | 12712271 | 12711702 |
Note that since , , , and are equinumerous,
(1)
| |||
(2)
|
are also equinumerous.
Erdős proved that there exist at least one prime of the form and at least one prime of the form between and for all .