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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. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
This formula reduces to [3] [4] = + = [() <]; that is, it tautologically defines as the smallest integer m for which the prime-counting function is at least n. This formula is also not efficient. This formula is also not efficient.
Short video visualizing the Prime Number Theorem. Prime formulas and Prime number theorem at MathWorld. How Many Primes Are There? Archived 2012-10-15 at the Wayback Machine and The Gaps between Primes by Chris Caldwell, University of Tennessee at Martin. Tables of prime-counting functions by Tomás Oliveira e Silva; Eberl, Manuel and Paulson ...
The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. [99]
Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. The prime number theorem then states that x / log x is a good approximation to π ( x ) , in the sense that the limit of the quotient of the two functions π ( x ) and x / log x as x increases without bound is 1:
Meissel already found that for k ≥ 3, P k (x, a) = 0 if a = π(x 1/3).He used the resulting equation for calculations of π(x) for big values of x. [1]Meissel calculated π(x) for values of x up to 10 9, but he narrowly missed the correct result for the biggest value of x.
He then obtained the main result of the paper, a formula for J(x), by comparing with ln(ζ(s)). Riemann then found a formula for the prime-counting function π ( x ) (which he calls F ( x )). He notes that his equation explains the fact that π ( x ) grows more slowly than the logarithmic integral , as had been found by Carl Friedrich Gauss and ...
where p n is the nth prime number. As n tends to infinity, R n is asymptotic to the 2nth prime, i.e., R n ~ p 2n (n → ∞). All these results were proved by Sondow (2009), [3] except for the upper bound R n < p 3n which was conjectured by him and proved by Laishram (2010). [4] The bound was improved by Sondow, Nicholson, and Noe (2011) [5] to