Pretentious distance between one and the Möbius function ($D(1,\mu ;X)\sim \sqrt{2\log \log X}$)

Because $\mu (p)=-1$, the squared pretentious distance from one to the Möbius function is twice the reciprocal-prime sum. The Mertens second theorem therefore gives $D(1,\mu ;X{)}^2=2\log \log X+2B_1+o(1)$. In particular the distance diverges, illustrating that a multiplicative function can be far from the constant function even though both have unit-modulus prime values.