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Modular Prime Counting Function


By way of analogy with the prime counting function pi(x), the notation pi_(a,b)(x) denotes the number of primes of the form ak+b less than or equal to x (Shanks 1993, pp. 21-22).

Hardy and Littlewood proved that pi_(4,1)(n) an pi_(4,3)(n) switches leads infinitely often, a result known as the prime quadratic effect. The bias of the sign of pi_(4,3)(n)-pi_(4,1)(n) is known as the Chebyshev bias.

Groups of equinumerous values of pi_(a,b) include (pi_(3,1), pi_(3,2)), (pi_(4,1), pi_(4,3)), (pi_(5,1), pi_(5,2), pi_(5,3), pi_(5,4)), (pi_(6,1), pi_(6,5)), (pi_(7,1), pi_(7,2), pi_(7,3), pi_(7,4), pi_(7,5), pi_(7,6)), (pi_(8,1), pi_(8,3), pi_(8,5), pi_(8,7)), (pi_(9,1), pi_(9,2), pi_(9,4), pi_(9,5), pi_(9,7), pi_(9,8)), and so on. The values of pi_(n,k) for small n are given in the following table for the first few powers of ten (Shanks 1993).

npi_(3,1)(n)pi_(3,2)(n)pi_(4,1)(n)pi_(4,3)(n)
SloaneA091115A091116A091098A091099
10^11212
10^211131113
10^380878087
10^4611617609619
10^54784480747834808
10^639231392663917539322
10^7332194332384332180332398
10^82880517288093728805042880950
10^925422713254248202542349125424042
npi_(6,1)(n)pi_(6,5)(n)
SloaneA091115A091119
10^111
10^21112
10^38086
10^4611616
10^547844806
10^63923139265
10^7332194332383
10^828805172880936
10^92542271325424819
npi_(7,1)(n)pi_(7,2)(n)pi_(7,3)(n)pi_(7,4)(n)pi_(7,5)(n)pi_(7,6)(n)
SloaneA091120A091121A091122A091123A091124A091125
10^1011010
10^2345354
10^3282730262927
10^4203203209202211200
10^5159315841613160116041596
10^6130631306513105130691310513090
10^7110653110771110815110776110787110776
10^8960023960114960213960085960379960640
10^9847422184747968475123847402184746308474742
npi_(8,1)(n)pi_(8,3)(n)pi_(8,5)(n)pi_(8,7)(n)
SloaneA091126A091127A091128A091129
10^10111
10^25766
10^337444343
10^4295311314308
10^52384240923992399
10^619552196531962319669
10^7165976166161166204166237
10^81439970144054414405341440406
10^912711220127123401271227112711702

Note that since pi_(8,1)(n), pi_(8,3)(n), pi_(8,5)(n), and pi_(8,7)(n) are equinumerous,

pi_(4,1)(n)=pi_(8,1)(n)+pi_(8,5)(n)
(1)
pi_(4,3)(n)=pi_(8,3)(n)+pi_(8,7)(n)
(2)

are also equinumerous.

Erdős proved that there exist at least one prime of the form 4k+1 and at least one prime of the form 4k+3 between n and 2n for all n>6.


See also

Chebyshev Bias, Dirichlet's Theorem, Prime Counting Function, Prime Quadratic Effect

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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. http://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."

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Modular Prime Counting Function

Cite this as:

Weisstein, Eric W. "Modular Prime Counting Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModularPrimeCountingFunction.html

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