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Prime counting function

Revision as of 12:22, 13 August 2015 by Pi3point14 (talk | contribs)

The prime counting function, denoted $\pi$, is a function defined on real numbers. The quantity $\pi(x)$ is defined as the number of positive prime numbers less than or equal to $x$. Gauss first conjectured that the prime number theorem $\leadsto$ $\frac{x}{log x}$, or equivalently, $\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{log x}} = 1$.

The function $\pi(x)$ is asymptotically equivalent to $\frac{x}{log x}$. This is the prime number theorem. It is also asymptotically equivalent to Chebyshev's theta function. It was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin, working independently.

See also

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