Argument inversion for modified theta functions (Q2772830)

Let \(\varphi(x) = \sum_{n=1}^\infty f(\pi x n^2)\), where \(f(x)=\frac{1}{e^x-1}\). The main aim of this article is to give another proof of the formula NEWLINE\[NEWLINE\varphi\left(\frac{1}{x}\right)-R\left(\frac{1}{x}\right) =\sqrt{\frac{x}{2}}\sum_{n\geq 1}\frac{\left\{\omega^{-1}f(2\pi\omega\sqrt{2nx})+\omega f(2\pi\omega^{-1}\sqrt{2nx})\right\}}{\sqrt{n}},NEWLINE\]NEWLINE where NEWLINE\[NEWLINER(x) = \frac{\pi}{6x}+\frac{\zeta\left(\frac{1}{2}\right)}{2\sqrt{x}}+\frac{1}{4},NEWLINE\]NEWLINE and \(\omega\) is a primitive eighth root of unity. The method used here works when \(n^2\) in the definition of \(\varphi(x)\) is replaced by \(n^{2k}\) for any positive integer \(k\).