Author:
(1) Yitang Zhang.
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
7. Mean-value formula I
Let N (d) denote the set of positive integers such that h ∈ N (d) if and only if every prime factor of h divides d (note that 1 ∈ N (d) for every d and N (1) = {1}). Assume 1 ≤ j ≤ 3 in what follows. Write
and
For notational simplicity we write
Let
with
For ψ(mod p) ∈ Ψ write
Let a = {a(n)} denote a sequence of complex numbers satisfying
Write
The goal of this section is to prove
In this and the next two sections we assume that 1 ≤ j ≤ 3.
Proof of Proposition 7.1: Initial steps
Here Proposition 2.1 is crucial.
Let κ(n) be given by
we obtain
By (7.4), the proof of (7.3) is reduced to showing that
This yields (7.5) by Proposition 2.1 and (2.9).
By (7.3) we may write
This yields
By trivial estimation, this remains valid if the constraint (l, p) = 1 is removed. Further, by the relation
we have
Thus the right side of (7.7) is
For (l, k) = 1 we have
Inserting this into (7.8) we deduce that
where
and
Proof of Proposition 7.1: The error term
In this subsection we prove (7.11).
Changing the order of summation gives
Assume 1 < r < D and θ is a primitive character (mod r). By Lemma 5.6, the right side of (7.14) is
which are henceforth assumed.
For σ = 1, by the large sieve inequality we have
It follows by Cauchy’s inequality that
This yields (7.15).
Proof of Proposition 7.1: The main term
In this subsection we prove (7.10).
Assume p ∼ P. We may write
The innermost sum is, by the Mellin transform, equal to
By the simple bounds
for σ > 9/10, we can move the contour of integration in (7.19) to the vertical segments
and to the two connecting horizontal segments
This yields
On the other hand, by Lemma 5.2 (ii) and direct calculation we have
Combining these with (7.20) and (7.21) we obtain (7.10), and complete the proof of Proposition 7.1.
This paper is available on arxiv under CC 4.0 license.