Just as the mathematicians below were all well aware, as the author of this paper, Xiao Yi understood even more clearly just how important Part I was to the entire work.
Therefore, he broke down even the smallest steps of this section and explained them in meticulous detail.
Time gradually passed, and the content became increasingly crucial.
Only his voice remained in the hall. Everyone else seemed to be attending a classical music concert, not even daring to cough.
"...Now, we begin by considering twisted sums over arithmetic progressions. When the modulus lies in a range D that is neither too small nor too large, we can obtain the result with the coefficient b replaced by an arbitrary complex number αrs."
"This means that for the range D, we can handle the sums S1 and S2 simultaneously, because the Jacobi symbol (s/r) can be absorbed into βz."
"If we factorize αrs, this would give a Barban-type result and follow the large sieve given earlier."
"So here, we need some new ideas."
"Automorphic theory of étale algebraic varieties."
At this point, Xiao Yi paused and made a joke. "Recently, a group known as the beneficiaries of automorphic theory of étale algebraic varieties has emerged in the mathematical community. And now, it seems that I have become one of them as well."
Laughter rose from the audience.
However, most of it came from those seated in the back, people who only half understood or did not understand at all.
As for the giants seated in the front rows, they merely forced faint smiles, inwardly thinking: Stop digressing. The most crucial step is next. Hurry up and explain it!
Xiao Yi did not disappoint them. He continued, "In the next step, we must express it in terms of Automorphic Forms. Then, we introduce