# Best error terms for functions related to square free numbers

Let $\mu$ denote the Möbius function, and let's define the functions $\mu_{-}$ and $\mu_{+}$ as follows: $\mu_{-}(n):=\frac{\mu(n)^{2}-\mu(n)}{2}$ and $\mu_{+}(n):=\frac{\mu(n)^{2}+\mu(n)}{2}$. Let $M_{-}$ be the summatory function of $\mu_{-}$ and $M_{+}$ the summatory function of $\mu_{+}$. One has $M(x)=M_{+}(x)-M_{-}(x)$, where $M$ is the summatory fonction of $\mu$, and $\hat{M}(x):=M_{+}(x)+M_{-}(x)$ is the number of squarefree integers below $x$. One has $\hat{M}(x)=\frac{1}{\zeta(2)}x+O(\sqrt{x})$. Let $\alpha$ and $\beta$ be such that $M_{-}(x)=\frac{1}{2\zeta(2)}x+O(x^{\alpha})$ and $M_{+}(x)=\frac{1}{2\zeta(2)}x+O(x^{\beta})$. If one can have $\alpha=\beta$ then these two quantities can be taken equal to $\frac{1}{2}$, and thus one would have $M(x)\ll\sqrt{x}$ which would imply RH. What are the best known (unconditional) error terms for $M_{+}$ and $M_{-}$? Is there any piece of evidence aside RH that they should be the same up to a multiplicative bounded quantity?
• You can read asymptotics for $M_{+}$ and $M_{-}$ directly from those for $M$ and $\hat{M}$. The error term for $M_{+}$ and $M_{-}$ are therefore $\frac{1}{2} M(x) + O(x^{1/2})$ and $-\frac{1}{2} M(x) + O(x^{1/2})$, respectively. Hence, the error terms have the same order of magnitude (unless $M(x) \ll \sqrt{x}$, which contradicts Gonek's conjecture). The best unconditional bound on $M(x)$ was given by Ivic and states that $$M(x) = O\left(x \exp\left(-c_{1} \log^{3/5} x (\log \log x)^{-1/5}\right)\right).$$ See the paper by Nathan Ng here. – Jeremy Rouse Aug 5 '14 at 17:51
As I say in the comment, the asymptotics for $M_{+}$ and $M_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M_{-}(x) = \frac{1}{2 \zeta(2)} x - \frac{1}{2} M(x) + O(x^{1/2})$. These error terms will have the same order of magnitude unless $M(x)$ is small.
If $M(x) \ll \sqrt{x}$ for all $x$, then Gonek's conjecture mentioned in the paper of Ng is false. However, there will probably be infinitely many $x$ for which $M(x)$ is small, and for these $x$ it will be less clear what the size of the error terms of $M_{+}$ and $M_{-}$ will be. (Corollary 1 of Ng's paper, which is very conditional, implies among other things that the logarithmic density of $x$ for which $M(x) \leq \epsilon \sqrt{x}$ tends to zero as $\epsilon \to 0$.)
Finally, the best unconditional bound on $M(x)$ was given by Ivic in his book "The Riemann Zeta-Function" and states that $$M(x) = O\left(x \exp\left(-c_{1} (\log^{3/5} x) (\log \log x)^{-1/5}\right)\right).$$ (The best conditional bound was given by Soundararajan in 2009.)