如何看待 arXiv2111.02792 对黎曼猜想的证明？ 第1页

撤了，做数学得和年轻人合作更有把握，毕竟很多时候检查细节是体力活，年龄不饶人。老头子精神可嘉，现在还能想数学就值得赞一个。国内这个年龄还能玩玩麻将就不错了。

搬运一下，方便大家参考。第八节部分讨论可以看看。

8. A logical paradox

A contradiction will originate from a comparison of Theorem 7.3 with

Theorem 8.1 below. We shall allow it to develop: then, we shall show that it would cease to hold in a mildly intuitionistic logic. That the Riemann hypothesis could thus arbitrate up to some point between positions in mathematical logic which were the subject of heated disputes a century ago certainly goes beyond our initial project. But we would be the last to reject the idea that R.H. could influence our understanding of the foundations of mathematics. Another, more trivial, explanation for the “paradox” to come could of course be that we made an error somewhere. In such a case, the present section may help readers kind enough to check my paper, or intent on finding an error in it: we thank them, whatever the results of their

investigations. But we rechecked our proof many times, though, of course,

we shall never consider our revision duties as completed.

We certainly did feel some uneasiness with the paradoxical results to be developed now. We felt encouraged to publicize these for the following reasons. First, there is no doubt in our mind that our methods, mixing analysis, algebra and arithmetic, provide a right direction. Next, Theorem 7.3 “explains” the reason why we shall never be able to produce a “bad” zero ρ, or to show that such a bad zero exists within a given range of imaginary values of ρ: this corroborates numerical evidence, pushed to an extremely high degree. In the other direction, Theorem 8.1 indicates that we cannot depend on the existence of δ < 1 such that all zeros have real parts ≤ δ: again, this fits with the experience gained by many mathematicians who, in the course of more than a century, made improvements in the construction of zero-free regions.