TOPICS

# 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 .

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

## Explore with Wolfram|Alpha

More things to try:

## 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 Web Resource. https://mathworld.wolfram.com/ModularPrimeCountingFunction.html