 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
The prime number theorem then states that
where ln(x) is the natural logarithm of x. This notation means only that the limit of the quotient of the two functions π(x) and x/ln(x) as x approaches infinity is 1; it does not mean that the limit of the difference of the two functions as x approaches infinity is zero.
An even better approximation, and an estimate of the error term, is given by the formula
for x → ∞ (see big O notation). Here Li(x) is the offset logarithmic integral function.
Here is a table that shows how the three functions (π(x), x/ln(x) and Li(x)) compare:
| x | π(x) | π(x) - x/ln(x) | Li(x) - π(x) | x/π(x) |
|---|---|---|---|---|
| 101 | 4 | 0 | 2 | 2.500 |
| 102 | 25 | 3 | 5 | 4.000 |
| 103 | 168 | 23 | 10 | 5.952 |
| 104 | 1,229 | 143 | 17 | 8.137 |
| 105 | 9,592 | 906 | 38 | 10.430 |
| 106 | 78,498 | 6,116 | 130 | 12.740 |
| 107 | 664,579 | 44,159 | 339 | 15.050 |
| 108 | 5,761,455 | 332,774 | 754 | 17.360 |
| 109 | 50,847,534 | 2,592,592 | 1,701 | 19.670 |
| 1010 | 455,052,511 | 20,758,029 | 3,104 | 21.980 |
| 1011 | 4,118,054,813 | 169,923,159 | 11,588 | 24.280 |
| 1012 | 37,607,912,018 | 1,416,705,193 | 38,263 | 26.590 |
| 1013 | 346,065,536,839 | 11,992,858,452 | 108,971 | 28.900 |
| 1014 | 3,204,941,750,802 | 102,838,308,636 | 314,890 | 31.200 |
| 1015 | 29,844,570,422,669 | 891,604,962,452 | 1,052,619 | 33.510 |
| 1016 | 279,238,341,033,925 | 7,804,289,844,392 | 3,214,632 | 35.810 |
| 4 ·1016 | 1,075,292,778,753,150 | 28,929,900,579,949 | 5,538,861 | 37.200 |
As a consequence of the prime number theorem, one get an asymptotic expression for the nth prime number p(n):
One can also derive the probability that a random number n is prime: 1/ln(n).
The theorem was conjectured by Adrien-Marie Legendre in 1798 and proved independently by Hadamard and de la Vallée Poussin in 1896. The proof used methods from complex analysis, specifically the Riemann zeta functionIn mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in physics. Definition The Riemann zeta function ζ s is.
Because of the connection between the Riemann zeta functionIn mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. It also has applications in physics. Definition The Riemann zeta function ζ s is and π(x), the Riemann hypothesisThe Riemann hypothesis first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ s . It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize ha has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.
Helge von KochNiels Fabian Helge von Koch ( January 25, 1870 March 11, 1924) was a Swedish mathematician, who gave his name to the famous fractal known as the Koch curve, which was one of the earliest fractal curves to have been described. He was born into a family of in 1901Events January 1 World celebrates what is regarded as the start of the new century. Zero-ists' argument that new century should be celebrated in 1900 rejected worldwide). January 1 The six colonies that make up Australia are federated as under an act of t showed that more specifically, if the Riemann hypothesis is true, the error term in the above relation can be improved toThe constant involved in the O-notation is unknown.
| Politics | Business | Finance |
 |
 |
 |
|