Can Riemann's Function Prove an Equation for Prime Numbers?

[jostpuur]

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

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

it satisfies an equation

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

Anyone knowing how to prove that?

 

[mathwonk]

well just glancing at it, doubling that function and adding one would appear to result from just summing over all integers n. see if that helps.
doesn't seem to.

or read harold edwards book studying this paper in detail.