On établit les majorations $\left|M\left(x\right)\right|\le \frac{0.002969x}{{(logx)}^{1/2}}$, valable pour $x\ge 142194,\left|M\left(x\right)\right|\le \frac{0.6437752x}{logx}$ qui est la meilleure majoration possible en $\frac{x}{logx}$ valable pour tout $x\>1(\left|M\left(5\right)\right|=2=\frac{0.6437752\times 5}{log5})$, et d’autres analogues. On montre enfin comment trouver des majorations effectives $\left|M\left(x\right)\right|\>\frac{c_{k}x{(loglogx)}^{2k}}{{(logx)}^k}$ pour tout $k$.

In a recent work we gave some estimations for exponential sums of the form ${\sum }_{n\le x}\Lambda \left(n\right)exp\left(2i\pi (f\left(n\right)+\beta n)\right)$, where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.