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