Riemann Hypothesis/The hypothesis

Theorem 1

Consider the functional equation for Zeta,

Notice that for , the sine term evaluates to which evaluates to 0 for all integers , hence for all natural .

Definition 1

These zeroes are referred to as trivial zeroes. As a set,

Zeroes that do not lie in this set are to be referred to as non-trivial zeroes.

Note 1

The above argument cannot be applied to , as is a simple pole (), as are negative arguments of ().

Theorem 2

All non-trivial zeroes of have a real part that lies in the interval

Theorem 3

Take the inequality,

Using the definition of deduced in an earlier chapter,

Taking the log of both sides, using

Writing as a power series,

Substituting ,

Taking the modulus of the argument,

Substituting appropriate values,

If one lets , it should become apparent that,

Clearly implying,

Exponentiating both sides,

Let's assume that has a zero at , then,

As gives a pole, and gives a zero, contradicting the previously stated inequality, proving theorem 3 by contradiction .

Theorem 4

Using the functional equation,

By theorem 3, the RHS is non-zero, hence as is the LHS.

Theorems 3 and 4 are sufficient to imply theorem 2.

The conjecture


Riemann, knowing that all zeroes lied in the critical strip, postulated,


All non-trivial zeroes of   have a real part of  

The above conjecture is considered to be the most notable in pure mathematics, and the most notable of Riemann's works.