Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

2 Jun 2024


(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


15. Evaluation of Φ1

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





First we prove that


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


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


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


This yields



On substituting n = mk we can writ



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


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.