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Peter Borwein (in his 2006 book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, p. 6) provides an equivalence between the Riemann Hypothesis and this conjecture involving the Liouville function:

The connections between the Liouville function and the Riemann Hypothesis were explored by Landau in his doctoral thesis of 1899.

Theorem 1.2. The Riemann Hypothesis is equivalent to the statement that for every fixed ε > 0, $$\displaystyle\lim_{n \to \infty}\frac{\lambda\left(1 \right)+ \lambda\left(2 \right) + ... + \lambda\left(n \right)}{n^{\frac{1}{2}+\epsilon}} =0$$

Borwein cites the theorem to Landau's 1899 doctoral thesis, but the proof doesn't seem to be in Michael J. Coons' English translation (http://arxiv.org/pdf/0803.3787v2.pdf).

These similar previous posts (Riemann Hypothesis, is this statement equivalent to Mertens function statement? and How do you prove that $M(N)=O(N^{1/2+\epsilon})$ from the Riemann Hypothesis?) lead me to believe that the proof has to do with the Mellin transform except instead on the Liouville function?

Would anyone mind explaining where the theorem is located in the thesis or where it was proven elsewhere?

Thanks.

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  • $\begingroup$ $\frac{1}{\zeta(s)} = s \int_1^\infty M(x) x^{-s-1} dx$ and $\frac{\zeta(2s)}{\zeta(s)} = s \int_1^\infty L(x) x^{-s-1} dx$ where $L(x) = \sum_{n \le x} \lambda(n)$. It follows that $L(x) = \sum_{k=1}^{\sqrt{x}} M(x/k^2)$ and $M(x) = \sum_{k=1}^{\sqrt{x}} \mu(k) L(x/k^2)$ so that $M(x)= \mathcal{O}(x^{1/2+\epsilon})$ iff $L(x) = \mathcal{O}(x^{1/2+\epsilon})$ $\endgroup$ – reuns Jul 16 '16 at 18:44

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