@helmut_meukelI do have a PhD in Mathematics so I will take this as my cue.
The description of how a mathematician thinks about a problem is fairly good, but the technical details are a jumble. (My principal objections relating to the thought processes are tiny ones: the time frame is absurdly short for the work she is doing; also, the "Eureka" moments usually come when you step away, not while you are staring at the problem.)
As you observe, "Lemma 9" is just whichever lemma is ninth in her paper as she wrote it. That won't match "Lemma 9" in any other paper. A lemma is just one step in a bigger proof. (In general, a lemma gets a name when the original lemma proves to be useful in other contexts so that it is referenced by other papers. Effectively, such a lemma is used as a theorem in its own right.)
The biggest mathematical problem is the translation issue. The story makes a big deal of the error in translation that causes the translated conjecture to differ from the original conjecture. To a mathematician, that just means that there are now two conjectures, both of which may be worth working on. (Incidentally, there are many first-rate German-speaking mathematicians, so it requires some suspension of disbelief that everybody has accepted the erroneous translation for 60 years.)
Apart from that, the story is correct to say that significant advances often come from combining branches of mathematics. However, in the story, she says she is using topology to solve problems in number theory, but then she appears to use graph theory instead of topology, and Lemma 9 seems to be stuck in real analysis.
Historically, both real and complex analysis have been used to solve problems in number theory. I could believe that the story is simplifying the situation for the benefit of laymen, and that really: the earlier papers had already established a framework for addressing the problem using real analysis, she was transforming the problem from real analysis into graph theory, then embedding her graph into some kind of manifold, and then using topological methods... Or, possibly, the story writes "interval" where the mathematicians are discussing topological neighbo(u)rhoods or open sets.
In any case, the concept of the open interval (the interval without its endpoints) is well established, and it is unlikely that any mathematician would spend time debating the necessity of including the endpoints (that is, using the closed interval instead).
On top of all that, there is a reference to Gödel, which is expected if she is proving that the conjecture is undecidable, or even unprovable. However, the visiting expert gives the halting problem as an example, which it is not. Then the protagonist responds that she is not proving that the conjecture is unprovable, but rather that it is provable. In general, the way to show that something is provable is to prove it, without bringing in Gödel's work. (I am not aware of any conjecture that has been shown to be provable without proving it.)
There are some other, non-mathematical problems, like having a three-member panel composed of Patterson, Jamison, Simpson, and Bennett. There is also no explanation of how she arranges to skip school for over a week.
Having said all of that, I still think it is a good story. Something that is probably not obvious to the layman is how she is solving the problem of her mother taking the bus: Proving the Riemann Conjecture would win her a prize of USD 1M.