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Understanding the Mean-Value Formula II with Primitive Charactersby@eigenvalue

Understanding the Mean-Value Formula II with Primitive Characters

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Eigen Value Equation Population

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June 2nd, 2024
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Mean-Value Formula II involves proving Proposition 14.1, using primitive characters and sequences of complex numbers to establish the result under the condition |β| < 5α.
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Eigen Value Equation Population

Eigen Value Equation Population

@eigenvalue

We cover research, technology, & documentation about special scalar values associated with square matrices. #EigenValue

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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

14. Mean-value formula II


Recall that we always assume ψ is a primitive character (mod p), p ∼ P. Sometimes we write pψ for the modulus p.


Let k ∗ = {κ ∗ (m)} and a ∗ = {a ∗ (n)} denote sequences of complex numbers satisfying


image


The goal of this section is to prove


Proposition 14.1. Suppose |β| < 5α. Then


image


image


image


image


This paper is available on arxiv under CC 4.0 license.


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We cover research, technology, & documentation about special scalar values associated with square matrices. #EigenValue

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