The Wikipedia article on the Chebyshev function $\psi(x)$ states that, evaluated at $x=e^t$, it minimizes the functional

$$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\infty \int_0^\infty e^{-st}f(s)f(t)ds dt,$$

so that $f(t) = \psi(e^t)e^{-ct}$ for $c>0$.

Wikipedia lists no source for this. Can someone please point me to a text proving this (and possibly more on the connection between analytic number theory and variational calculus)?


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