T2. 1. Introduction

Question 1

State prime number theorem

Answer 1

π(x) = #{p ≤ x: p is prime} ~ x/logx as x→∞

Question 2

What does it mean for f(x) ~ g(x)

Answer 2

lim x→∞ f(x)/g(x) = 1

Question 3

State Dirichlet’s theorem

Answer 3

For any coprime positive integers a,q there are infinitely many primes p ≡ a (modq)

Question 4

State Euler’s identity

Under what condition does this converge?

Answer 4

SUM 1,∞ 1/n^s = PROD_p 1/1-p^-s

Re(s) > 1

Question 5

Define the Riemann-zeta inequality

Answer 5

1/s-1 < ζ(s) < 1/s-1 + 1

Question 6

Give a brief understanding of the infinitude of primes

Answer 6

Consdier the ζ(s) inequalities. These indicate that the function diverges as s→1, where as each term in the product rep remains bounded.