Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty e^{-\pi k^2 t}=\frac{1}{2}(\Theta(t)-1)$$ has also a functional equation.
QUESTION:
Does somebody know, what functional equation does $$T(t):=\sum\limits_{k=1}^\infty ke^{-\pi k^2 t}$$ have ?
