Reciprocity for The Short Twisted Second Moment of The Riemann Zeta Function

We prove a reciprocity formula for a twisted second moment of the Riemann zeta function over a short interval of length $T^{\delta }$, where $\delta \in (\frac{1}{2},1)$, centered at height $T$ on the critical line. This result extends Khan’s reciprocity formula to the short interval setting and makes explicit the relation between the twisting parameters and the interval length.