Summing the Möbius and Liouville functions \[M(x)=2M(\sqrt{x})-\sum_{j\leq\sqrt{x}}\mu(j)\sum_{k\leq\sqrt{x}}\mu(k)\left\lfloor{\frac{x}{jk}}\right\rfloor\] \[M(x)=M(\sqrt{x})-\sum_{\scriptsize j\leq\sqrt{x}}\mu(j)\left(\sum_{\scriptsize\frac{\sqrt{x}}{j}<k\leq\sqrt{\frac{x}{j}}}M\left(\frac{x}{jk}\right)+\sum_{\scriptsize k\leq\sqrt{\frac{x}{j}}}\mu(k)\left\lfloor{\frac{x}{jk}}\right\rfloor-M\left(\scriptsize\sqrt{\frac{x}{j}}\right)\left\lfloor{\scriptsize\sqrt{\frac{x}{j}}}\right\rfloor\right)\] \[L(x)=L(\sqrt{x})-\sum_{j\leq\sqrt{x}}\mu(j)\left(\sum_{k\leq\sqrt{x}}\lambda(k)\left\lfloor{\frac{x}{jk}}\right\rfloor-\left\lfloor{\scriptsize\sqrt{\frac{x}{j}}}\right\rfloor\right)\] \[L(x)=-\sum_{\scriptsize j\leq\sqrt{x}}\mu(j)\left(\sum_{\scriptsize\frac{\sqrt{x}}{j}<k\leq\sqrt{\frac{x}{j}}}L\left(\frac{x}{jk}\right)+\sum_{\scriptsize k\leq\sqrt{\frac{x}{j}}}\lambda(k)\left\lfloor{\frac{x}{jk}}\right\rfloor-\left(L\left(\scriptsize\sqrt{\frac{x}{j}}\right)+1\right)\left\lfloor{\scriptsize\sqrt{\frac{x}{j}}}\right\rfloor\right)\]