Number Theory: An Introduction to Mathematics

366 IX The Number of Prime Numbers

Conversely, supposepn/nlogn→1. Since

(n+ 1 )log(n+ 1 )/nlogn=( 1 + 1 /n){ 1 +log( 1 + 1 /n)/logn}→ 1 ,

it follows thatpn+ 1 /pn→1. Furthermore

logpn−logn−log logn→ 0 ,

and hence

logpn/logn→ 1.

Ifpn≤x<pn+ 1 ,thenπ(x)=nand

nlogpn/pn+ 1 ≤π(x)logx/x≤nlogpn+ 1 /pn.

Since

nlogpn/pn+ 1 =pn/pn+ 1 ·nlogn/pn·logpn/logn→ 1

and similarlynlogpn+ 1 /pn→1, it follows that alsoπ(x)logx/x→1.

Numerical evidence, both for the primenumber theorem and for the fact thatLi(x)
is a better approximation thanx/logxtoπ(x), is provided by Table 1.
In a second paper Chebyshev (1852) made some progress towards proving the
prime number theorem by showing that

a≤ lim

x→∞

π(x)logx/x≤ lim

x→∞

π(x)logx/x≤ 6 a/ 5 ,

wherea= 0 .92129. He used his results to give the first proof ofBertrand’s postulate:
for every realx>1, there is a prime betweenxand 2x.
New ideas were introduced by Riemann (1859), who linked the asymptotic behav-
iour ofπ(x)with the behaviour of the function

ζ(s)=

∑∞

n= 1

1 /ns

Table 1.

x π(x) x/logxLi(x)π(x)logx/xπ(x)/Li(x)

103 168 144. 177. 1.16 0.94

104 1 229 1 085. 1 245. 1.132 0.987

105 9 592 8 685. 9 629. 1.1043 0.9961

106 78 498 72 382. 78 627. 1.08449 0.99835

107 664 579 620 420. 664 917. 1.07117 0.99949

108 5 761 455 5 428 681. 5 762 208. 1.06130 0.99987

109 50 847 534 48 254 942. 50 849 234. 1.05373 0.999966

1010 455 052 511 434 294 481. 455 055 614. 1.04780 0.999993