Difference between revisions of "Prime counting function"
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| − | The '''prime counting function''', denoted <math>\pi</math>, is a [[function]] defined on [[real number]]s. The quantity <math>\pi(x)</math> is defined as the number of [[positive]] [[prime number]]s less than or equal to <math>x</math>. Gauss first conjectured that the [[prime number theorem]] <math>\leadsto</math> <math>\frac{x}{log x}</math>, or equivalently, <math>\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{log x}}</math>. | + | The '''prime counting function''', denoted <math>\pi</math>, is a [[function]] defined on [[real number]]s. The quantity <math>\pi(x)</math> is defined as the number of [[positive]] [[prime number]]s less than or equal to <math>x</math>. Gauss first conjectured that the [[prime number theorem]] <math>\leadsto</math> <math>\frac{x}{log x}</math>, or equivalently, <math>\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{log x}} = 1</math>. |
The function <math>\pi(x)</math> is [[asymptotically equivalent]] to <math>\frac{x}{log x}</math>. 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. | The function <math>\pi(x)</math> is [[asymptotically equivalent]] to <math>\frac{x}{log x}</math>. 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. | ||
Revision as of 12:22, 13 August 2015
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, 
.
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
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