On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity

Consider the Dirichlet polynomial and its prime-indexed sub-sum $$\begin{array}{ccc}E(t):={\sum }_{k\le \left[\sqrt{\frac{t}{2\pi }}\right]}\frac{e^{it\log k}}{\sqrt{k}},& ;& E_1(t):={\sum }_{\begin{array}{c}p\le \left[\sqrt{\frac{t}{2\pi }}\right]\\ p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\end{array}}\frac{e^{it\log p}}{\sqrt{p}}\mathrm{.}\end{array}$$ In this work we investigate the statistical and arithmetic properties of the sets $${\mathcal{C}}_N:=\{E(g_n):1\le n\le N\},{\mathcal{P}}_N:=\{E_1(g_n):1\le n\le N\},$$ obtained by sampling these sums at the Gram points $g_n$ of the Hardy $Z$-function. We refer to ${\mathcal{C}}_N$ and ${\mathcal{P}}_N$ as the full and prime $N$-th Gram clouds, respectively. The motivation is that the approximate functional equation expresses both $Z(g_n)$ and $Z^{\mathrm{\prime }}(g_n)$ through different projections of the same oscillatory phase configuration, suggesting that the geometry of these sampled sums may retain information about the local behavior of nearby zeros. Numerically, we find that Gram-point sampling produces highly non-generic geometric structures: good and isolated bad Gram points occupy markedly different regions of the clouds, the prime cloud of isolated bad points splits into two anisotropic lobes, and the position along the principal direction of the folded prime cloud is strongly correlated with the viscosity of the Hardy $Z$-function $\mu (g_n)=\mid \frac{Z^{\mathrm{\prime }}(g_n)}{Z(g_n)}\mid ,$ with smallest-viscosity points concentrating near distinguished extremal regions of the lobes.