$$\large I(s)= \int_1^\infty\psi(x)(x^{\frac{1-s}{2}}-x^{\frac{\overline{s}}{2}})\frac{dx}{x}$$ where $s\in \mathbb{C}$ and $\psi (x)=\sum_{n=1}^\infty e^{-n^2\pi x}$ , $\overline{s}$ denotes the complex conjugate
Prove that $I(s)=0$ when $0<\Re(s)<1$
My try:
$I(1/2+it)=\int_1^\infty\psi(x)(x^{\frac{1/2-it}{2}}-x^{\frac{1/2-it}{2}})\frac{dx}{x}$
$I(1/2+it)=0 \ \forall \ t\in \mathbb{R}$
So, $I(s)=0$ for infinitely many values of $s$