Assuming that I understood correctly one claim from the Riemann's On the Number of Prime Numbers less than a Given Quantity, then if we define a function(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\psi:]0,\infty[\to\mathbb{R},\quad \psi(x) = \sum_{n=1}^{\infty} e^{-n^2\pi x},

[/tex]

it satisfies an equation

[tex]

2\psi(x) + 1 = x^{-\frac{1}{2}}\Big(2\psi\big(\frac{1}{x}\big) + 1\Big).

[/tex]

Anyone knowing how to prove that?

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# A detail from Riemann's paper

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