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8. Miscellaneous

    1. Problem 8.1.

      [T. Wooley] Can one calculate moments of zeta closer to the $\sigma=1$ line instead of on the $\sigma=\frac{1}{2}$ line? Can one also identify the lower order terms in the $\sigma=1$ case?
        1. Remark. [K. Soundararajan] In the papers of Conrey and Keating, the heuristics work for any values of the $\alpha$ and $\beta$ shift variables. They make the shifts tend to $0$ but may not need to.
            • Problem 8.2.

              [M. Radziwill] Develop a heuristic for twisted moments, e.g. $$ \int_0^T \left( \frac{m}{n}\right)^{it} |\zeta(\tfrac{1}{2}+it)|^{2k}\,dt. $$
                • Problem 8.3.

                  [A. Harper] Develop a heuristic for $$ \int_T^{2T} \left| \sum_{p\leq X} \frac{1}{p^{1/2+it}}\right|^{2k} \,dt. $$ Find connections with the work of Bogomolny-Keating and with the Ratios Conjectures.
                    1. Remark. [Adam Harper] A key issue here is to take $k$ large. Unless $X^{k} \geq T$ we don’t need a heuristic anyway, we can just apply mean value results for Dirichlet polynomials. But when I suggested the problem I had in mind very large $k$ (e.g. growing with $T$ at some rate), for which this would connect with questions about the maximum size of the zeta function.
                        • Problem 8.4.

                          [H. Iwaniec] Evaluate moments of zeta over sets other than $[0,T]$ or $[T,2T]$.
                            • Problem 8.5.

                              [H. Iwaniec] Evaluate $$\int_0^T f(|\zeta(\tfrac{1}{2}+it)|)\,dt,$$ where $f=J_{2k}$, the Bessel function, or $f=T_{2k}$, the Chebyshev polynomial.
                                • Problem 8.6.

                                  [H. Iwaniec] Evaluate $$ \sum \int_{t_j}^{t_j+T^{7/8}} |\zeta(\tfrac{1}{2}+it)|^4 \,dt, $$ where the sum is over a set of $t_j$’s with modulus $\leq T$ and spaced more than $T^{7/8}$ apart from each other (related to work of Heath-Brown and of Zavorotnyi)
                                    • Problem 8.7.

                                      [S. Lester] Evaluate fractional moments of zeta.
                                        1. Remark. [M. Rubinstein] There are conjectures on lower order terms, but there is no analogue yet of Conrey-Keating
                                            • Problem 8.8.

                                              [B. Conrey] Extend the work of Goldston and Gonek on mean value theorems for long Dirichlet polynomials by using Type II sums.

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