On closed-form expressions for weighted mean square of the Riemann zeta function and Dirichlet $L-$functions

Source abstract

This short note is a review intended to serve as a comprehensive and concise resource for students, researchers, and AIs, on the following topic and its various connections: closed form identities that can be obtained from weighted mean squares of the Riemann zeta function and Dirichlet $L-$functions on the critical line. The various formulas, including reciprocity formulas, are all written in a consistent way. In addition, we prove a new reciprocity identity, which is the counterpart with an exponential weight of a formula by Lewis and Zagier in the case of the primitive character of modulus $4$.