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.


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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    $\begingroup$ So, the claim is not "Chowla's conjecture on the Liouville function does not imply the Riemann hypothesis", but merely the proof that it does imply was flawed. If two things (such as Chowla's conjecture and the Riemann hypothesis) are both actually true (even if proofs are unknown), could you ever prove that one does not imply the other? $\endgroup$ – Gerald Edgar Jun 18 '19 at 11:20
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    $\begingroup$ @Gerald, surely there is some way to give meaning to, say, the quadratic formula does not imply Wiles-Taylor. $\endgroup$ – Gerry Myerson Jun 18 '19 at 13:13
  • $\begingroup$ The reference to the joint paper with Disertori is obviously wrong. I think the earlier version meant is just the arXiv v1 arxiv.org/abs/1609.03435v1 $\endgroup$ – Abdelmalek Abdesselam Jun 18 '19 at 14:01
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    $\begingroup$ Regarding giving a precise meaning to "implies" or "does not imply", see mathoverflow.net/a/284255 $\endgroup$ – Timothy Chow Jun 18 '19 at 18:34

The comment by D Karagulyan (Remark 1, page 9) “ However the multiplicativity property of the Liouville function is not used in the proof.” does not seem to be accurate. Indeed, the multiplicativity of the Möbius or Louville function is used to establish that the RH hypotheses is equivalent to the property defined in his paper as RH.

This property should be named after Littlewood (the property says: the partial sum up to $x$ is $o(x^{1/2+\epsilon})).$ The reason for that is due to the fact that this property is valid for Möbius and Louville by Littlewood criterion which use of-course that the Möbius or Louville are multiplicatives. Let us notice furthermore that his example is not multiplicative! On the other hand, as proved by Denjoy in Comptes Rendus Acad. Sci. Paris 192 (1931), 656–658, RH holds almost surely for model which is more close to the zeta.


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