Removal of the factorization assumption on the modulus

Establish the asymptotic formula for the mixed second moment of the Dirichlet L-function and the twisted holomorphic GL(2) L-function over moduli q without assuming that q admits the suitable factorization q=q_1q_2 used in the paper.

Background

The main theorem evaluates the mixed moment over moduli possessing a prescribed factorization q=q_1q_2, with q_2 restricted to a short range. The factorization is essential in the paper's treatment of the off-diagonal terms, particularly in the application of the hyper-Kloosterman and p-adic stationary-phase arguments.

The authors explicitly identify removal of this structural restriction as a problem for future research. Solving it would extend the theorem from the specially factored family of moduli to general moduli.

References

In our case, due to certain technical difficulties, we have to assume that $q$ also admits a suitable factorization. Removing this assumption would be an interesting problem for future research.

— The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point  (2608.25812 - Chen et al., 26 Aug 2026) in Section 1, Introduction