= Large Fourier coefficient criterion for pretentiousness
{title2=$|\widehat\nu(t)|\leq C\log X\,e^{-D(f,n^{it};X)^2}$}
Let $X\geq3$, and let $f$ be a normalized <multiplicative arithmetic function> supported on <squarefree integers>, with $|f(p)|\leq1$. Form the <finite measure> $\nu=\sum_n f(n)n^{-1-1/\log X}\delta_{\log n}$ using <Dirac measures>. Under the transform convention $\widehat\nu(t)=\int e^{-itu}\,d\nu(u)$, its <Euler product> is $\prod_p(1+f(p)p^{-1-1/\log X-it})$. Expanding its logarithmic modulus, and replacing the damped <prime> weights by $1/p$ up to $X$ with <Mertens first theorem>, gives the displayed absolute bound, uniformly in real $t$. Hence $|\widehat\nu(t)|\geq\delta\log X$ forces $D(f,n^{it};X)^2\leq\log(C/\delta)$. The distance expression still defines a nonnegative pretentious score for <prime> values bounded by one; its actual <metric> interpretation uses unit-modulus values.
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