Modular Prime Counting Function

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	A091115	A091119
	1	1
	11	12
	80	86
	611	616
	4784	4806
	39231	39265
	332194	332383
	2880517	2880936
	25422713	25424819

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 .

REFERENCES:

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, p. 96, 2004.

Granville, A. and Martin, G. "Prime Number Races." Aug. 24, 2004. https://www.arxiv.org/abs/math.NT/0408319.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.

Sloane, N. J. A. Sequences A073505, A073506, A073508, A091098 A091099, A091115, A091116, A091117, A091119, A091120, A091121, A091122, A091123, A091124, and A091125 in "The On-Line Encyclopedia of Integer Sequences."