Prime counting function

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

The function is asymptotically equivalent to . 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

• Prime Number Theorem
• Riemann zeta function

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