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Deriving an Approximate Formula for Dirichlet L-Functionsby@eigenvalue

Deriving an Approximate Formula for Dirichlet L-Functions

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Eigen Value Equation Population

@eigenvalue

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June 2nd, 2024
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This section derives an approximate formula for L(s, ψ), leveraging the functional equation and contour integration, providing a detailed analysis with error estimation and key lemmas.
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Eigen Value Equation Population

Eigen Value Equation Population

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

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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

6. Approximate formula for L(s, ψ)

Write


image


Let


image


Lemma 6.1. Suppose ψ(mod p) ∈ Ψ, |σ − 1/2| < 2α and |t − 2πt0| < L1 + 2. Then


L(s, ψ) = K(s, ψ) + Z(s, ψ)N(1 − s, ψ¯) + O(E1(s, ψ)),


where


image


and where


image


Proof. By (4.3) we have


image


The left side above is, by moving the line of integration to u = −1, equal to


image


It therefore suffices to show that


image


For u = −1 we have, by the functional equation (2.2) with θ = ψ,


image


We first show that


image


We move the contour of integration in (6.2) to the vertical segments


image


and


image


with the horizontal connecting segments


image


image


whence (6.2) follows. The proof of (6.1) is therefore reduced to showing that


image


This paper is available on arxiv under CC 4.0 license.


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