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?

ispossible (Rudin-Shapiro polynomials, etc.). So, I'm extremely skeptical but going into all details at midnight is not what I want to try. $\endgroup$ – fedja Jun 18 at 4:54