The twelfth moment of Dirichlet $L$-functions with smooth moduli

Heath-Brown proved a strong boung for the twelfth moment of Riemann zeta function that has the nice feature of giving the Weyl's estimate as a corollary. It is very natural to ask similar questions for Dirichlet $L$-functions as we let the moduli $q$ tend to infinity, the so-called $q$-aspect. The exact analog of Heath-Brown's result for general moduli is not known. Indeed it would improve upon the celebrated Burgess' bound for Dirichlet $L$-functions. An easier path is to restrict one's attention to numbers having a nice factorization. There are two orthogonal ways of doing so. We can either consider numbers that are large powers of a fixed prime or consider moduli $q$ that are squarefree and all its prime factors are smaller than a small power of $q$. In this talk we will take the latter route and show how one can prove the exact analog of Heath-Brown's result in this context.