WOLFRAM|DEMONSTRATIONS PROJECT

Extending Rosser's Theorem

range

20

variation k

1

Let

π(x)

be the number of primes up to

x

. The prime number theorem states that

lim

x∞

π(x)

x/ln(x)

=1

and implies that

p

n

~nln(n)

, where

p

n

is the

th

n

prime. Rosser proved that

p

n

>nln(n)

for all

n=1,2,3,…

. Rosser's theorem was extended to

p

n

>n(ln(n)+ln(ln(n))-1)

, for all

n>1

.

The curves plotted are

p(n)

(blue),

n(ln(n)+ln(ln(n))-k)

(khaki), and

nln(n)

(brown).