# Are the Erdos $L^4$ conjecture and related Turyn-Golay conjecture proved?

I apologise if this is off-topic.

There is a two year old paper see here on arXiv claiming a solution to these conjectures on norms of polynomials with real coefficients of magnitude 1.

Erdos' $$L^4$$ conjecture:

Let $$P(z)=\frac{1}{\sqrt{q}}\left( \epsilon_0 +\epsilon_1 z +\cdots +\epsilon_{q-1} z^{q-1} \right),\quad \epsilon_i=\pm 1,$$ with $$z\in S^1.$$ Then, $$\lVert P \rVert_4>(1+c),$$ where $$c>0.$$

The paper claims to prove Erdos' $$L^4$$ conjecture which then implies the Turyn-Golay conjecture that the merit factor of any binary sequence is bounded. Which in turn implies that there are only finitely many Barker sequences with perfect aperiodic autocorrelation.

These conjectures are related to the Littlewood $$L^1$$ conjecture proved by McGehee-Pigno and Smith.

Not being an expert, while the arguments look plausible to me, I was wondering if any of the experts here are aware of this paper, or have an opinion on its claims.

Edit: The following has been moved to a separate question, see here since it can be independently answered:

The author also claims that Chowla's conjecture (applied to the Liouville function instead of the Mobius function), i.e., that} $$\lim_{N\rightarrow \infty} \sum_{n\leq N} \lambda(n+a_1) \lambda(n+a_2) \cdots \lambda(n+a_k)=o(N),$$ implies the Riemann hypothesis. Is this claim new?

• The last paragraph ("Is this claim new?") looks like a reasonable question to me, but the general request for "an opinion on its claims" may fall afoul of the sentiments expressed here: meta.mathoverflow.net/questions/927/… See also meta.mathoverflow.net/a/2332/3106 – Timothy Chow Jun 6 at 14:27
• The passage "Otherwise H mod r is almost equipped with the uniform probability measure of {0,···,r−1}, which also yields a contradiction." followed by the "end of proof" sign on page 24 doesn't look too promising. As far as I can tell, the author considers various trivial reasons for each of which he (or she?) concludes that the $L^4$-norm tends to $+\infty$. That is all fine, but the interesting case is when the $L^4$ norm does not blow up and that is possible (Rudin-Shapiro polynomials, etc.). So, I'm extremely skeptical but going into all details at midnight is not what I want to try. – fedja Jun 18 at 4:54
• Thanks @fedja, that's very helpful. Perhaps you can have a look later, when it's not midnight where you are :-) – kodlu Jun 18 at 5:05
• @kodlu Maybe. Meanwhile you can make the judgement yourself. Ignore all the blah-blah-blah about how one great conjecture implies another one and just read the "proof of the first main result" skipping the detailed considerations of the cases that are obvious to you. In 2-3 hours at most you'll come to a definite conclusion. :-) – fedja Jun 18 at 17:16