Mathematicians revive abandoned approach to the Riemann Hypothesis

By coincidence I toyed with the problem a bit this past week. Pretty difficult to tackle with a "basic" math toolbox (calc, trig, algebra, and some discrete math/number theory that's taught in many CS undergrad curricula). Much like Fermat's Last Theorem, it seems like this might require much higher-level math to solve what seems like a simple problem.

The basic idea is to prove that a certain function L(N) doesn't grow too quickly. It's true that appropriate bounds on L(N) imply RH.

But Eswaran's arguments for this point are full of handwaving, and in this field handwaving is just not good enough. L(N) is a sum of terms +1 and -1, and he claims that it behaves like a series of coin tosses, but the specific ways he argues it's like a series of coin tosses are not enough for the conclusion he needs.

E.g., in section 5, on page 13, he states a theorem which says: you can pair up the +1 and -1 terms in that series, so that for each +1 there's a -1 and for each -1 there's a +1. This is certainly true, but it doesn't imply that the series is random-like or that the partial sums L(N) don't grow much faster than sqrt(N) (which is the condition he needs). E.g., consider the series that goes ++-++-++-++-++-++- etc., whose partial sums obviously grow like N/3. This satisfies that "pairing" condition just fine. (Pair the first + with the first -, the second + with the second -, etc. There are infinitely many of both, which is all you need to make that work.)

In appendix V, he tries a different approach. (The fact that he gives two different "proofs" of the same key result, and both are handwavy, is a Very Bad Sign.) There's lots here that looks wrong, but here's one thing: he claims to prove that if |L(N)| ~ N^a for some a then we must have a=1/2, and having given his proof he claims to have proved that in fact |L(N)| ~ N^1/2. But these are very different propositions. Another possibility is that we don't have |L(N)| ~ N^a for any a at all, which in fact is true because it's known that infinitely often |L(N)|=sqrt(N).

The lectures appear to be about his alleged proof rather than about RH in general. In any case, I would not trust someone capable of making the errors in his RH paper to teach any research-level material in pure mathematics.

The world would be a better place if more people had the same approach to problem-solving, IMHO.