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Extending Rosser's Theorem

range

variation k

Let

π(x)

be the number of primes up to

. The prime number theorem states that

lim

x∞

π(x)

x/ln(x)

and implies that

~nln(n)

, where

is the

prime. Rosser proved that

>nln(n)

for all

n=1,2,3,…

. Rosser's theorem was extended to

>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).

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