The prime number theorem then states that
where ln(x) is the natural logarithm of x. This notation means 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) 

10^{1}  4  0  2  2.500 
10^{2}  25  3  5  4.000 
10^{3}  168  23  10  5.952 
10^{4}  1,229  143  17  8.137 
10^{5}  9,592  906  38  10.430 
10^{6}  78,498  6,116  130  12.740 
10^{7}  664,579  44,159  339  15.050 
10^{8}  5,761,455  332,774  754  17.360 
10^{9}  50,847,534  2,592,592  1,701  19.670 
10^{10}  455,052,511  20,758,029  3,104  21.980 
10^{11}  4,118,054,813  169,923,159  11,588  24.280 
10^{12}  37,607,912,018  1,416,705,193  38,263  26.590 
10^{13}  346,065,536,839  11,992,858,452  108,971  28.900 
10^{14}  3,204,941,750,802  102,838,308,636  314,890  31.200 
10^{15}  29,844,570,422,669  891,604,962,452  1,052,619  33.510 
10^{16}  279,238,341,033,925  7,804,289,844,392  3,214,632  35.810 
4 ·10^{16}  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):
The theorem was conjectured by AdrienMarie 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 function. Nowadays, socalled "elementary" proofs are available that only use number theoretic means. The first of these was provided partly independently by Paul Erdös and Atle Selberg in 1949 although it was prior believed that such proofs with only real variables can not be found.
Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis 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 Koch in 1901 showed that more specifically, if the Riemann hypothesis is true, the error term in the above relation can be improved to
The constant involved in the Onotation is unknown.
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