A famous theorem of Selberg asserts that $\log|\zeta(\tfrac12+it)|$ is approximately a normal distribution with mean $0$ and variance $\tfrac12\log\log T$, when we sample $t\in [T,2T]$ uniformly. This extends in a natural way to a plethora of other $L$-functions, one of them being Dirichlet $L$-functions $L(s,\chi)$ with $\chi$ a primitive Dirichlet character. Viewing $\zeta(\tfrac12+it)$ and $L(\tfrac12+it,\chi)$ as normal variables, we expect indepedence between them, meaning that for fixed $V_1,V_2 \in \mathbb{R}$: $$\textrm{meas}_{t \in [T,2T]} \left\{\frac{\log|\zeta(\tfrac12+it)|}{\sqrt{\tfrac12 \log\log T}}\geq V_1 \text{   and   } \frac{\log|L(\tfrac12+it,\chi)|}{\sqrt{\tfrac12 \log\log T}}\geq V_2\right\} \sim \prod_{j=1}^2 \int_{V_j}^\infty e^{-x^2/2} \frac{\textrm{d}x}{\sqrt{2\pi}}.$$
    When $V_j\asymp \sqrt{\log\log T}$, i.e. we are considering values of order of the variance, the asymptotic above breaks down, but the Gaussian behaviour is still believed to hold to order. For such $V_j$ the behaviour of the joint distribution is decided by the moments $$I_{k,\ell}(T)=\int_T^{2T} |\zeta(\tfrac12+it)|^{2k}|L(\tfrac12+it,\chi)|^{2\ell}\, dt.$$ We establish that $I_{k,\ell}(T)\asymp T(\log T)^{k^2+\ell^2}$ for $0<k,\ell \leq 1$. The lower bound holds for all $k,\ell >0$. This allows us to decide the order of the joint distribution when $V_j =\alpha_j\sqrt{\log\log T}$ for $\alpha_j \in (0,\sqrt{2}]$. Other corollaries include sharp moment bounds for Dedekind zeta functions of quadratic number fields, and Hurwitz zeta functions with rational parameter. 
    

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.