User:Darcourse/Books/Sieve of Eratosthenes to Riemann's Hypothesis

A proof of Riemann's Hypothesis

 * Prime beginnings
 * Sieve of Eratosthenes


 * Analytic Number Theory
 * Analytic number theory
 * Proof of the Euler product formula for the Riemann zeta function
 * Möbius function
 * Dirichlet series
 * Von Mangoldt function
 * Prime-counting function


 * Prime Number Theorem
 * Prime number theorem
 * Chebyshev function
 * Logarithmic integral function
 * Mertens' theorems


 * Towards Riemann's Hypothesis
 * Gamma function
 * On the Number of Primes Less Than a Given Magnitude
 * Riemann zeta function


 * Proving Rieman's Hypothesis
 * Proof by contradiction
 * Ratio test
 * Radius of convergence
 * Riemann hypothesis

Proof
If $$0.5+\epsilon$$ is a zero, then so is $$0.5+n\epsilon$$, $$n$$ an integer, by the ratio test. Therefore, unless $$\epsilon=0$$, $$0.5+n\epsilon>1$$ for some $$n$$, which contradicts no zeroes right of the $$1$$-line.