The zeros on the critical line
2.1 The infinity of zeros on the critical line
The distribution of zeros of ζ(s) on the critical line is one of the central topics in zeta-function theory. The aim of this section is to prove that ζ(s) has an infinity of zeros on the critical line. This assertion is equivalent to showing that Z(t) has arbitrarily large zeros in absolute value and, since Z(t) is even, we may consider only positive zeros. Suppose on the contrary that, for T ≥ T0, the function Z(t) does not change sign. Then
But it is not difficult ...