# Extending Rosser's Theorem

Extending Rosser's Theorem

Let be the number of primes up to . The prime number theorem states that =1 and implies that ~nln(n), where is the prime. Rosser proved that >nln(n) for all . Rosser's theorem was extended to , for all .

π(x)

x

lim

x∞

π(x)

x/ln(x)

p

n

p

n

th

n

p

n

n=1,2,3,…

p

n

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

n>1

The curves plotted are (blue), (khaki), and (brown).

p(n)

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

nln(n)