Introduction to Analytic Number Theory
《Introduction to Analytic Number Theory》 · 第 91 页 · Section 4.9「The partial sums of the Mobius function」
If $x>1$ we define $M(x)=\sum_{n<x}\mu(n)$. Numerical evidence suggests that $|M(x)|<\sqrt{x}$ if $x>1$, but this inequality, known as Mertens' conjecture, has not been proved nor disproved.