paint-brush
Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transformsby@eigenvalue
114 reads

Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

tldt arrow

Too Long; Didn't Read

The evaluation of Φ1 involves advanced mathematical techniques, including Mellin transforms and key lemmas, to derive precise results under specific conditions and propositions
featured image - Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms
Eigen Value Equation Population  HackerNoon profile picture

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



with



Write




where



Hence



First we prove that



Since




it follows that



This yields (15.4).


Let κ1(m) be given by



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



On the other hand, we can write



with



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



where



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



where



This yields



Hence



where



On substituting n = mk we can writ



with



Hence



it follows that



If (q, dl) = 1, then



so that



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



It follows that the function



is analytic and it satisfies



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




The following lemma will be proved in Appendix B.



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



This yields, by (15.21),



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


Lemma 15.2. If |s − 1| < 5α, then



Lemma 15.3. For σ ≥ 9/10 the function



is analytic and bounded. Further we have



By (4.2) and (4.3),



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



This together with Lemma 15.2 and 15.3 yields



since



It follows by (15.22) that



By Lemma 5.8,



Hence, by direct calculation,



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



This paper is available on arxiv under CC 4.0 license.