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

# Evaluating Φ1: Integrating Lemmas, Propositions, and Mellin Transforms

June 2nd, 2024

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

Author:

(1) Yitang Zhang.

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.

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