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The Theory of Hardy’s Z-Function by Aleksandar Ivi

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2

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 images 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 TT0, the function Z(t) does not change sign. Then

images

But it is not difficult ...

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