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## 3. Number Theory and Zeta

1. #### Problem 3.1.

Is the value distribution of $|\zeta(1/2+it)|$ related to Painlevé?
• #### 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 $E_N(0,s)=\sum_{n=0}^N\frac{(-1)^n}{n!}\int_{\left[-\frac{s}{2},\frac{s}{2}\right]^n}\det_{n\times n}S_N(x_i-x_j)dx_1\dots dx_n$ 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.
Reference: Keating et al. "Sums of divisor functions in $F_ {q}[t]$ and matrix integrals." arXiv preprint arXiv:1504.07804 (2015).
• #### 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 off-diagonal 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.