3. Number Theory and Zeta

Problem 3.2.
Where is the arithmetic part in $P(s)$, the spacing distribution between zeros of the zeta function, for $\zeta(1/2+it)$? Can arithmetic corrections be found? 
We denote by $E_N(0,s)$ the measure of the set of matrices in $U(N)$ with no normalised eigenangle within $(0,s)$. Then we have that \[E_N(0,s)=\sum_{n=0}^N\frac{(1)^n}{n!}\int_{\left[\frac{s}{2},\frac{s}{2}\right]^n}\det_{n\times n}S_N(x_ix_j)dx_1\dots dx_n\] where $S_N(\theta)=\frac{\sin\frac{N\theta}{2}}{\sin\frac{\theta}{2}}$.
Problem 3.3.
Find the small $s$ expansion for $E_N(0,s)$  coefficients are polynomials in $N$. Find this expansion for all number theoretic ensembles in the literature. 
Recall that \[\zeta^k(s)=\sum_{n=1}^\infty\frac{d_k(n)}{n^s},\] and consider the truncated sum $\sum_{n=1}^N\frac{d_k(n)}{n^s}$. Then compute \[\lim_{T\rightarrow\infty}\frac{1}{(\log T)^{k^2}T}\int_0^T\left\sum_{n=1}^{T^\alpha}\frac{d_k(n)}{n^{1/2+it}}\right^2\frac{dt}{a_k}\overset{?}{=}\text{Piecewise Poly}_{k^2}(\alpha), \] where the equality is conjectured. Further, Keating et al showed that \[\mathbb{E}\left\sum_{j_1+\dots+j_k=j}Sc_{j_1}(t)\dots Sc_{j_k}(t)\right^2=\text{Piecewise Poly}(j/N).\] The $SC_i$ above are Secular coefficients.
Problem 3.4.
Prove the connection between these two equations  it is claimed that one is derivative of the other. 
Problem 3.5.
Is there a formulation of the Riemann Hypothesis in Function Fields in terms of Hankel determinants? 
Problem 3.6.
Within random matrix theory, how does one distinguish the diagonal and offdiagonal contributions (from the point of view of Number Theory)? Is there a statistic one can write down?
Cite this as: AimPL: Painleve equations and their applications, available at http://aimpl.org/painleveapp.