Estimate for partial sums of a series equivalent to the Riemann hypothesis

Question

The sums $$S_N=\sum_{n=1}^N\frac{\mu(n)}{n},$$ where $\mu$ is the Moebius function, are known to tend to 0 as $N\to+\infty$. As far as I remember, there was an estimate on $S_N$ equivalent to the Riemann Hypothesis. If so, could anyone please remind me the exact formulation and give the reference?

Also, what is the best known estimate?

Answer 1

Define $M(N)=\sum_{n\leq N}\mu(n)$. Then $M(N)\ll N^{1/2+\varepsilon}$ is equivalent to the Riemann Hypothesis (Aleksandar Ivic, The Riemann Zeta-Function, page 47). Define $S(N)=\sum_{n\leq N}\frac{\mu(n)}{n}$. We will prove

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$M(N)\ll N^{1/2+\varepsilon}$ if and only if $S(N)\ll N^{-1/2+\varepsilon}$.

Proof: Suppose $M(N)\ll N^{1/2+\varepsilon}$. Then by partial summation $$ \begin{align*} S(N)&=\frac{1}{N}\sum_{n\leq N}\mu(n)+\int_1^N\frac{1}{t^2}\sum_{n\leq t}\mu(n)\,d t\\ &=\frac{M(N)}{N}+\int_1^N\frac{M(t)}{t^2}\,dt\\ &\ll N^{-1/2+\epsilon} \end{align*} $$ Suppose $S(N)\ll N^{-1/2+\varepsilon}$. Again, by partial summation $$ \begin{align*} M(N)=\sum_{n\leq N}\frac{\mu(n)}{n}n&=N\sum_{n\leq N}\frac{\mu(n)}{n}-\int_1^N\sum_{n\leq t}\frac{\mu(n)}{n}\,d t\\ &\ll NS(N)+\left|\int_1^N S(t)\,dt\right|\\ &\ll N^{1/2+\epsilon}. \end{align*} $$ This Completes the proof

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