MathOverflow

Let $n$ be a positive integer.
Is it possible to choose a finite set of proper divisors$$D=\{d_1,d_2,\dots,d_k\}, \qquad 1<d_i<n,\ d_i\mid n,$$such that$$\sum_{i=1}^k d_i = n,$$and such that $D$ is an antichain under divisibility, i.e. for any $i\neq j$,$$d_i \nmid d_j \quad...

I am trying to justify a Tauberian/singularity-transfer step for a Dirichlet series whose analytic continuation has a logarithmic-type singular expansion at $s=1$. Concretely, I have an analytic function $\Phi(s)$ that coincides with a Dirichlet series...

This is a follow up question from this answer. Let:
$F$ be a field of characteristic $\mathfrak c$ (zero or positive);
$G=\{g_1(=1),g_2,\dots,g_n\}\le F^\times$ (hence $|G|=n$);

Let $f \in L^2(0,1)$ and $T>0$ be fixed. How can I choose $g \in L^2(0,T)$ such that\begin{align*}0\equiv \mathbb{E}^x\left[f\left(X_T\right) \chi_{\tau \geqslant T}+g(T-\tau) \chi_{X_\tau=1}\right] \ \ ?\end{align*}Here $\chi$ stands for the characteristic...