8. Miscellaneous

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?
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 BogomolnyKeating and with the Ratios Conjectures.
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 HeathBrown and of Zavorotnyi) 
Problem 8.7.
[S. Lester] Evaluate fractional moments of zeta.
Remark. [M. Rubinstein] There are conjectures on lower order terms, but there is no analogue yet of ConreyKeating


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.