MathOverflow

This is a reference request. Are there any results of the following kind?
Assume $\mathrm{CH}$ and let $\mathcal{U}$ be a Ramsey ultrafilter. Let $c$ be a Cohen real. Then in $V[c]$, can $\mathcal{U}\cup\{c\}$ be extended to a Ramsey ultrafilter again? Instead of a Cohen real, any results concerning other splitting reals would...

Background.Robin’s theorem (1984) states that the Riemann Hypothesis is equivalent to the inequality
σ(n)<eγnlog⁡log⁡nfor alln
5040,σ(n)<eγnloglognfor all n>5040,whereσ(n)σ(n) is the sum-of-divisors function andγγ is the Euler–Mascheroni constant.​It is also known that any least counterexample, if it exists, must lie among certain extremal sequences such as superabundant...

Are there references that compute the integral motivic cohomology $\mathrm{H}^{p,q}(BG,\mathbb Z)$, for $G$ finite cyclic and where $BG$ is defined over the rationals.
I am particularly interested in $G=\mu_2$.
Thanks in advance!

$\def\C{\mathrm{C}}\def\P{\mathrm{P}}\def\AR{\mathrm{AR}}$(All rings are commutative and unital.) I am trying to prove or disprove the following:
($\C$). Iterated adic completion amounts to joint adic completion. More precisely: Let $R$ be a ring and $a_1,\dots,a_n\in R$. Suppose $M$ is an $R$-module. Define $M_0=M$ and $M_i=M^{\wedge a_i}_{i-1}$, $i=1,\dots,n$. Then $M_n\cong M^{\wedge\mathfrak...