Prime counting function
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In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by
- π(x)
and may be expressed formally as
- <math>\pi(x) = \sum_{p \le x,\ p\ \mathrm{prime}} 1.<math>
Of great interest in number theory is the growth rate of this function. It was conjectured in the 18th century to be approximately
- <math> x/\operatorname{ln}(x)<math>.
What is now called the prime number theorem is one precise estimate for this rate of growth, namely that
- <math>\pi(x) / \operatorname{li}(x)<math> approaches 1
as <math> x<math> increases without bound, where li is the logarithmic integral function. This was first proved around 1896 by Jacques Hadamard and by de la Vallée Poussin (independently), using properties of the zeta function introduced by Bernhard Riemann in 1859.
More precise estimates of <math>\pi(x)<math> are now known; for example
- <math>\pi(x) = \operatorname{li}(x) + O(x \exp(-\frac{\sqrt{\ln(x)}}{15}))<math>,
where the O is Big O notation. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdös (for the most part independently).
Other prime counting functions
Other prime counting functions are also used because they are more convenient to work with. One is Riemann's prime counting function, denoted <math>\Pi(x)<math> or <math>J(x)<math>. This has jumps of 1/n for prime powers pn, with it taking a value half-way between the two sides at discontinuities. That added detail is because then it may be defined by an inverse Mellin transform. Formally, we may define J by
- <math>J(x) = \frac12 \left(\sum_{p^n < x} \frac1n\ + \sum_{p^n \le x} \frac1n\right)<math>
where p is a prime.
We may also write
- <math>J(x) = \sum_{n=1}^\infty \pi(x^\frac{1}{n})<math>
except where we have discontinuities at prime powers, and hence π can be recovered from J by Möbius inversion.
Another prime counting function weights prime powers pn by ln p:
- <math>\psi(x) = \frac12 \left(\sum_{p^n < x} \ln p\ + \sum_{p^n \le x} \ln p\right).<math>
Formulas for prime counting functions
These come in two kinds, arithmetic formulas and analytic formulas. The latter are what allow us to prove the prime number theorem. From the work of Riemann and von Mangoldt, we have the following expression for J:
- <math>J(x) = \operatorname{Li}(x) - \sum_{\rho}\operatorname{Li}(x^{\rho}) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t} <math>
Here Li is the offset logarithmic integral, and the ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest, and the sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part.
For ψ we have a simpler formula, due to von Mangoldt:
- <math>\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} + \ln \frac{x^2}{x^2-1} - \ln {2 \pi}.<math>
Again, the formula is valid for x > 1.
The Riemann hypothesis
The Riemann hypothesis is equivalent to a much sharper bound on the error in the estimate for <math>\pi(x)<math>, and hence to a more regular distribution of prime numbers,
- <math>\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log{x}).<math>