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