Now that a week has passed since Zhang posted his preprint Discrete mean estimates and the Landau–Siegel zero on the arXiv, I'm wondering if someone can give a high-level overview of his strategy. In more detail: the paper first claims to show that an abnormally small value for the $L$-function at $s=1$ (or equivalently, a zero close to $s=1$), will influence the zeros of other $L$-functions, forcing them to lie on the critical line (which is expected) and to be very regularly spaced (which definitely is not expected). This part of the paper has a long history, going under the name of the Deuring–Heilbronn phenomenon. Many researchers claimed results in this direction in conference papers, including Montgomery (Zhang's reference [17]) and Heath-Brown (Zhang's [13]). Full proofs were obtained by Conrey and Iwaniec (‘Spacing of zeros of Hecke L-functions and the class number problem’, Acta Arith. 103 (2002), no. 3, 259–312). Mark Watkins subsequently published ‘Comments on Deuring's zero-spacing phenomenon’, J. Number Theory 218 (2021), 1–43.
As far as I can see, what Zhang claims in this part is similar to the Conrey–Iwaniec result, lower bounding the $L$-function value by a power of logarithm under a hypothesis on the spacing of zeros. (If I've missed something, let me know.)
The rest of Zhang's paper claims to show that this regular spacing does not happen. Ruling out this ‘Alternative Hypothesis’ has been a goal in analytic number theory for decades, and despite the work of many (Montgomery for test functions with restricted support, Katz–Sarnak for the function field analog, Odlyzko and Rubinstein on numerical data for the zeros of the zeta function and Dirichlet $L$-functions, respectively), no one has been able to rule out this regular spacing. (For specific values of the discriminant, specific spacing of numerically computed zeros has had applications since Stark's 1964 UC Berkeley PhD thesis "On the tenth complex quadratic field with class-number one"
Zhang gives only a few hints of the strategy. In the abstract, “by evaluating certain discrete means of the large sieve type, a contradiction can be obtained….”
At the bottom of page 8 and top of page 9 “This is analogous to the results of Conrey, Iwaniec, and Soundararajan [5] and [6]. However, it seems that such a choice … is not good enough for our purpose. … instead, a variant of the argument will be adopted.”
What is it he is trying to do?
