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8. Miscellaneous


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      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.


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              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.


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                  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.


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                          Problem 8.4.

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


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                              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.


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                                  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)


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                                      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


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                                              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.