MathOverflow

I am interested in the following Boolean-ring question, motivated by a renorming problem for Banach spaces of density $\omega_1$.
Let $\mathfrak A$ be a Boolean ring, not necessarily with a maximal element. I shall write $a\leq b$ when $ab=a$. For $a,i\in\mathfrak A$, put...

Fix an integer $k \geq 2$ and let $n \geq 2$ be integer as well. Let $\lambda_1,\lambda_2 \in \{-1, 1 \}$, and then consider the set$$S_{k}^{\lambda_1,\lambda_2}:= \{n \in \mathbb{N}\setminus\{0,1\} : \gcd(n^k+\lambda_1, \hspace{2mm} n!+\lambda_2) &...

Let $X$ be a connected scheme. Remark 4.2.14 from Bhatt-Scholze "The pro-etale topology for schemes" states that the constant sheaf functor $\mathop{\rm Set}\to \mathop{\rm Shv}(X_{et})$ is not always limit-preserving. Is there an example of the failure...

I have been working on finding exact, closed-form discrete evaluations for a specific family of combinatorial sums, aiming to find exact identities where standard asymptotic or continuous approximations are typically used.
Through this work, I have found an identity. For $a > 1$, $b \le c$, and $d \ge 0$:...