= Order-one growth of the Riemann xi function
{title2=$\log M_\xi(R)\asymp R\log R$}
The <pole-subtracted theta integral for the completed zeta function> bounds the <Riemann xi function> on $|s|\leq R$ by a polynomial in $R$ times $\int_1^\infty e^{-\pi t}t^{(R+1)/2}\,dt$. The <Stirling formula> gives logarithmic growth $O(R\log R)$. On the positive real axis, the definition and $\zeta(R)\geq1$ give $\log\xi(R)=\frac R2\log R+O(R)$. Thus its <order of an entire function> is exactly one.
// Target: analytic-number-theory.bigb
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