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5. Other families of L-functions


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      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?


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          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.


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                  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).


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                      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?


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                              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?


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                                  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?


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                                      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.