= Prime-restriction metric for pretentious distance
{title2=$D(f,g;X)^2=\tfrac12\sum_{p\leq X}|f(p)-g(p)|^2/p$}
For unit-modulus <prime> values, <pretentious distance> is the Euclidean <metric> on the finite vectors $(f(p)/\sqrt{2p})_{p\leq X}$. On whole <arithmetic functions> it is only a <pseudometric>: it sees neither values on <prime powers> nor <primes> above the cutoff. Even for every cutoff, the functions one and $\mu^2$ have distance zero but differ on nonsquarefree integers. The <norm> representation proves the <triangle inequality> and identifies exactly the quotient on which separation holds.
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