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Deriving Mean-Value Formula I for Dirichlet L-Functionsby@eigenvalue

Deriving Mean-Value Formula I for Dirichlet L-Functions

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Eigen Value Equation Population

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June 2nd, 2024
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This section delves into the proof steps for Proposition 7.1, focusing on the analysis of error terms, main terms, and the application of tools like the large sieve inequality and the Mellin transform in deriving Mean-Value Formula I for Dirichlet L-functions.
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STORY’S CREDIBILITY

Academic Research Paper

Academic Research Paper

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

Author:

(1) Yitang Zhang.

  1. Abstract & Introduction
  2. Notation and outline of the proof
  3. The set Ψ1
  4. Zeros of L(s, ψ)L(s, χψ) in Ω
  5. Some analytic lemmas
  6. Approximate formula for L(s, ψ)
  7. Mean value formula I
  8. Evaluation of Ξ11
  9. Evaluation of Ξ12
  10. Proof of Proposition 2.4
  11. Proof of Proposition 2.6
  12. Evaluation of Ξ15
  13. Approximation to Ξ14
  14. Mean value formula II
  15. Evaluation of Φ1
  16. Evaluation of Φ2
  17. Evaluation of Φ3
  18. 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


image


and


image


For notational simplicity we write


image


Let


image


with


image


For ψ(mod p) ∈ Ψ write


image


Let a = {a(n)} denote a sequence of complex numbers satisfying


image


Write


image


The goal of this section is to prove


image


image


In this and the next two sections we assume that 1 ≤ j ≤ 3.


Proof of Proposition 7.1: Initial steps


image


Here Proposition 2.1 is crucial.


Let κ(n) be given by


image


image


we obtain


image


By (7.4), the proof of (7.3) is reduced to showing that


image


This yields (7.5) by Proposition 2.1 and (2.9).


By (7.3) we may write


image


This yields


image


By trivial estimation, this remains valid if the constraint (l, p) = 1 is removed. Further, by the relation


image


we have


image


Thus the right side of (7.7) is


image


For (l, k) = 1 we have


image


Inserting this into (7.8) we deduce that


image


where


image


and


image


Proof of Proposition 7.1: The error term


In this subsection we prove (7.11).


Changing the order of summation gives


image


image


Assume 1 < r < D and θ is a primitive character (mod r). By Lemma 5.6, the right side of (7.14) is


image


which are henceforth assumed.


image


For σ = 1, by the large sieve inequality we have


image


It follows by Cauchy’s inequality that


image


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


image


The innermost sum is, by the Mellin transform, equal to


image


By the simple bounds


image


for σ > 9/10, we can move the contour of integration in (7.19) to the vertical segments


image


and to the two connecting horizontal segments


image


This yields


image


image


On the other hand, by Lemma 5.2 (ii) and direct calculation we have


image


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.


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