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Author:
(1) Yitang Zhang.
Appendix A. Some Euler products
Appendix B. Some arithmetic sums
Write
Let
Lemma 6.1. Suppose ψ(mod p) ∈ Ψ, |σ − 1/2| < 2α and |t − 2πt0| < L1 + 2. Then
L(s, ψ) = K(s, ψ) + Z(s, ψ)N(1 − s, ψ¯) + O(E1(s, ψ)),
where
and where
Proof. By (4.3) we have
The left side above is, by moving the line of integration to u = −1, equal to
It therefore suffices to show that
For u = −1 we have, by the functional equation (2.2) with θ = ψ,
We first show that
We move the contour of integration in (6.2) to the vertical segments
and
with the horizontal connecting segments
whence (6.2) follows. The proof of (6.1) is therefore reduced to showing that
This paper is available on arxiv under CC 4.0 license.