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## Outline of the Proof

The proof of Theorem 1.1 involves many remarkable phenomena, and we wish to explain its structure in this chapter. To this end, we will give the first of a succession of statements, each of which implies the theorem. Passing from each statement to the next is a nontrivial reduction that changes the nature of the problem to be solved. We will outline the ideas of these reductions here and tackle them in detail in subsequent chapters.

Statement A. We have HΓ = HΔ.

This reduction was already mentioned in the first chapter, where Statement A appeared as Theorem 1.2.

The proof that this implies Theorem 1.1 is Theorem 1 of [15]. We review the idea of the proof. To prove the functional equations that ZΨ (s; m) is to ...

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