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Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transformsby@eigenvalue
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Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

by Eigen Value Equation Population
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Eigen Value Equation Population

@eigenvalue

We cover research, technology, & documentation about special scalar values...

June 2nd, 2024
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The evaluation of Φ1 involves advanced mathematical techniques, including Mellin transforms and key lemmas, to derive precise results under specific conditions and propositions
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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

15. Evaluation of Φ1

Recall that Φ1 is given by (13.8). In view of (12.2), B(s, ψ) can be written as


image


with


image


Write


image


image


where


image


Hence


image


First we prove that


image


Since



image


it follows that


image


This yields (15.4).


Let κ1(m) be given by


image


Regarding b as an arithmetic function, for σ > 1 we have


image


On the other hand, we can write


image


with


image


It follows by (15.3)-(15.5) and Proposition 14.1 that


image


where


image


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


image


where


image


This yields


image


Hence


image


where


image


On substituting n = mk we can writ


image


with


image


Hence


image


it follows that


image


If (q, dl) = 1, then


image


so that


image


for σ > 9/10. In case (q, dl) > 1 and σ > 9/10, the left side above is trivially


image


It follows that the function


image


is analytic and it satisfies


image


for σ > 9/10. The right side of (15.14) can be rewritten as


image


image


The following lemma will be proved in Appendix B.


image


By (15.19)-(15.21) and Lemma 15.1 we obtain


image


This yields, by (15.21),


image


To apply (15.22) we need two lemmas which will be proved in Appendix A.


Lemma 15.2. If |s − 1| < 5α, then


image


Lemma 15.3. For σ ≥ 9/10 the function


image


is analytic and bounded. Further we have


image


By (4.2) and (4.3),


image


By Lemma 15.3, we can move the contour of integration in the same way as in the proof of Lemma 8.4 to obtain


image


This together with Lemma 15.2 and 15.3 yields


image


since


image


It follows by (15.22) that


image


By Lemma 5.8,


image


Hence, by direct calculation,


image


Combining these relations with (15.23) , (15.17) and (15.6) we conclude


image


This paper is available on arxiv under CC 4.0 license.


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