Modular Prime Counting Function

The modular prime counting function counts the primes with , where and are integers. It is analogous to the prime counting function (Shanks 1993, pp. 21-22).

Littlewood proved that and change leads infinitely often (Granville and Martin 2004), a result known as the prime quadratic effect. The bias toward positive values of is known as the Chebyshev bias.

For fixed , the counts for residue classes relatively prime to are asymptotically equal: the ratio of any two tends to 1 as (Granville and Martin 2004). This does not imply equality of their values at finite . The following table gives for several small moduli and 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

Partitioning the residue classes modulo 4 into those modulo 8 gives the exact identities

			(1)
			(2)

Erdős proved that there exists at least one prime of the form and at least one prime of the form between and for all .