- 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,
Taking the modulus of the argument,
Substituting appropriate values,
If one lets , it should become apparent that,
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.