
## 5. Other families of $L$-functions

1. #### Problem 5.1.

[B. Conrey] Using the ideas of Conrey and Keating, develop a heuristic for conjecturing asymptotic formulas for high moments of quadratic Dirichlet $L$-functions. What new terms contribute to the main term when $k\geq 3$?
• #### Problem 5.2.

[B. Conrey and H. Iwaniec] Develop a heuristic for moments of $L$-functions of automorphic forms.
1. Remark. [B. Conrey] One of the problems here is that $\sum a_m a_{m+h}=0$, so there are no Type I terms.
• #### Problem 5.3.

[C. Turnage-Butterbaugh] Develop a heuristic for moments of imprimitive $L$-functions, or if your $L$-function factors into degree $1$ factors, e.g. $$\int |L(\tfrac{1}{2}+it,\chi_1)\cdots L(\tfrac{1}{2}+it,\chi_k)|^2\,dt$$ (or different powers).
• #### Problem 5.4.

[M. Milinovich] Develop a heuristic for quadratic twists of $L$-functions
1. Remark. [H. Iwaniec] What role does the root number play?
• #### Problem 5.5.

[H. Iwaniec and K. Soundararajan] The mechanism for computing moments in families like $\chi$ (mod $q$), $q\leq Q$, involve different orthogonality relations than $|m-n|=$small. Yet the answers are the same. Why?
• #### Problem 5.6.

[M. Radziwill] Is there any way to get lower bounds for small moments (e.g. first moment) in families, even assuming GRH?
• #### Problem 5.7.

[O. Balkanova] Find an exact formula for $\sum_{f\in H_k}^h L(\frac{1}{2},Sym^2f)^2$.

Cite this as: AimPL: Moments of zeta and correlations of divisor sums, available at http://aimpl.org/zetamoments.