Chowla’s Conjecture – Implications for the Riemann Hypothesis

analytic-number-theorynt.number-theory

A paper see here on arXiv 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. I have been unable to find any references to this claim after some research.

Is this claim new? Any pointers, references appreciated.

Best Answer

D Karagulyan, On certain aspects of the Mobius randomness principle, writes (Remark 1, page 9), "We remark, that the result proved above contradicts with what is claimed in [Reference 1]. There it is stated, that for the Liouville function the Chowla conjecture implies the Riemann hypothesis. However the multiplicativity property of the Liouville function is not used in the proof. But this can not be true as from the above argument it follows, that without the multiplicativity condition the Riemann hypothesis can not be obtained from the Chowla property."

Here [Reference 1] is E. H. el Abdalaoui, On the Erdos flat polynomials problem, Chowla conjecture and Riemann Hypothesis, https://arxiv.org/abs/1305.4361, so, an earlier version of the paper under discussion.

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