TOPICS
Search

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).

n pi_(3,1)(n) pi_(3,2)(n) pi_(4,1)(n) pi_(4,3)(n)
Sloane A091115 A091116 A091098 A091099
10^1 1 2 1 2
10^2 11 13 11 13
10^3 80 87 80 87
10^4 611 617 609 619
10^5 4784 4807 4783 4808
10^6 39231 39266 39175 39322
10^7 332194 332384 332180 332398
10^8 2880517 2880937 2880504 2880950
10^9 25422713 25424820 25423491 25424042
n pi_(6,1)(n) pi_(6,5)(n)
Sloane A091115 A091119
10^1 1 1
10^2 11 12
10^3 80 86
10^4 611 616
10^5 4784 4806
10^6 39231 39265
10^7 332194 332383
10^8 2880517 2880936
10^9 25422713 25424819
n pi_(7,1)(n) pi_(7,2)(n) pi_(7,3)(n) pi_(7,4)(n) pi_(7,5)(n) pi_(7,6)(n)
Sloane A091120 A091121 A091122 A091123 A091124 A091125
10^1 0 1 1 0 1 0
10^2 3 4 5 3 5 4
10^3 28 27 30 26 29 27
10^4 203 203 209 202 211 200
10^5 1593 1584 1613 1601 1604 1596
10^6 13063 13065 13105 13069 13105 13090
10^7 110653 110771 110815 110776 110787 110776
10^8 960023 960114 960213 960085 960379 960640
10^9 8474221 8474796 8475123 8474021 8474630 8474742
n pi_(8,1)(n) pi_(8,3)(n) pi_(8,5)(n) pi_(8,7)(n)
Sloane A091126 A091127 A091128 A091129
10^1 0 1 1 1
10^2 5 7 6 6
10^3 37 44 43 43
10^4 295 311 314 308
10^5 2384 2409 2399 2399
10^6 19552 19653 19623 19669
10^7 165976 166161 166204 166237
10^8 1439970 1440544 1440534 1440406
10^9 12711220 12712340 12712271 12711702

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

Explore with Wolfram|Alpha

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."

Referenced on Wolfram|Alpha

Modular Prime Counting Function

Cite this as:

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

Subject classifications

AltStyle によって変換されたページ (->オリジナル) /