Page 202 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

converges to a limit P , then the infinite product converges to the same limit. The Prime Number Theorem

was proven by understanding the Riemann zeta function, and in particular, for which complex numbers
z is ζ(z) =0. The Riemann hypothesis is a more precise statement about the complex numbers z for
which ζ(z) =0 which no mathematician has been able to prove or disprove.

Exercise: Read about the Riemann Hypothesis, the Riemann zeta function, the Gamma function, and
the Prime Number Theorem.

7.6 Exercises

1. Let
∞
{x n } n=1

be a sequence of positive numbers. If the sequence converges to a limit L , prove that

L ≥ 0.

∞ be a sequence of real numbers which converges to a limit L . If k ∈ N , prove
2. Let {x n } n=1
that the sequence
∞ ,
{y n } n=1 y n = x n+k

also converges to L.

202