For $k \in \mathbb{N}$ define $$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$ $$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$ By the p …

I would like to know if there is any formula for $ \prod_{x<p\leq y}\left(1+\frac1p\right) $ in terms of $\theta$ or $\psi$ functions $ \theta(x)=\sum_{p\leq x}\log p $ and $ …

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences. As …

By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form: $$\int \prod_ …

Let $$f : X_0 \rightarrow Y_0, \;\;\; g:X_1 \rightarrow Y_1$$ and define that the "direct product" of $f$ and $g$ is a map $$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times …

Let $s_1, s_2 \in (1/2,1\rbrack$. I would like to bound the product $$A=\prod_p \left(1 + \frac{p^{-s_1} p^{-s_2}}{(1-p^{-s_1}+p^{-1}) (1-p^{-s_2}+p^{-1})}\right)$$ Now, I am al …

Do disjoint unions and fiber products commute? In other words, is the following statement true? Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Le …

If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map $$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots …