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Author:
(1) Yitang Zhang.
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Table of Links
Abstract & Introduction
Notation and outline of the proof
The set Ψ1
Zeros of L(s, ψ)L(s, χψ) in Ω
Some analytic lemmas
Approximate formula for L(s, ψ)
Mean value formula I
Evaluation of Ξ11
Evaluation of Ξ12
Proof of Proposition 2.4
Proof of Proposition 2.6
Evaluation of Ξ15
Approximation to Ξ14
Mean value formula II
Evaluation of Φ1
Evaluation of Φ2
Evaluation of Φ3
Proof of Proposition 2.5
Appendix A. Some Euler products
Appendix B. Some arithmetic sums
15. Evaluation of Φ1
Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as
with
Write
where
Hence
First we prove that
Since
it follows that
This yields (15.4).
Let κ1(m) be given by
Regarding b as an arithmetic function, for σ > 1 we have
On the other hand, we can write
with
It follows by (15.3)-(15.5) and Proposition 14.1 that
where
The innermost sum above is, by the Mellin transform, equal to
where
This yields
Hence
where
On substituting n = mk we can writ
with
Hence
it follows that
If (q, dl) = 1, then
so that
for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially
It follows that the function
is analytic and it satisfies
for σ > 9/10. The right side of (15.14) can be rewritten as
The following lemma will be proved in Appendix B.
By (15.19)-(15.21) and Lemma 15.1 we obtain
This yields, by (15.21),
To apply (15.22) we need two lemmas which will be proved in Appendix A.
Lemma 15.2. If |s − 1| < 5α, then
Lemma 15.3. For σ ≥ 9/10 the function
is analytic and bounded. Further we have
By (4.2) and (4.3),
By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain
This together with Lemma 15.2 and 15.3 yields
since
It follows by (15.22) that
By Lemma 5.8,
Hence, by direct calculation,
Combining these relations with (15.23) , (15.17) and (15.6) we conclude
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This paper is available on arxiv under CC 4.0 license.
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