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We cover research, technology, & documentation about special scalar values associated with square matrices. #EigenValue
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Author:
(1) Yitang Zhang.
Appendix A. Some Euler products
Appendix B. Some arithmetic sums
We first prove a general result as follows.
By Proposition 7.1, our goal is reduced to evaluating the sum
Write
so that
Lemma 8.2. Suppose T < x < P. Then for µ = 6, 7
where
Proof. The sum is equal to
We move the contour of integration to the vertical segments
and to the two connecting horizontal segments
It follows by Lemma 5.6 that
The result now follows by direct calculation.
Combining these results with Lemma 8.3, we find that the integral (8.9) is equal to
The result now follows by direct calculation.
This paper is available on arxiv under CC 4.0 license.