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The $p$-adic logarithm can be computed $$ \begin{align*} \log_p(2) &= \frac{1}{p-1}\log_p(2^{p-1})\\ &=\frac{1}{p-1}\sum_{n\geq 1}(-1)^{n-1} \frac{(2^{p-1}-1)^n}{n} \\ &\equiv -pF(p)\mod p^2. \end{ali …

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where …

If $k$ is a number field, each embedding $\sigma:k\hookrightarrow\mathbb{C}$ determines a Weil cohomology theory $H^*_{B,\sigma}$ on smooth projective $k$-varieties given by taking the topological coh …

Not an answer, but an argument that your sum is between $(1/4+o(1))p\log^2(p)$ and $(3/4+o(1))p\log^2(p)$. Write $S(p)$ for your sum. Separate the sum into pieces according to the integer part of $(a …

I am considering the following two isomorphisms: First, if $X$ is a reasonably nice topological space, then $X$ has a normal covering space which is maximal with respect to the property of having an …

Let $n$ be a positive integer, $a_1,\ldots,a_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$, $$ \tag{$\star$} \sum_{i=1}^{n-1} i a_{(ui\!\!\!\mod n)} = 0. $$ Then th …

The matrix entry $*$ is $\log(u^{1-p})/p$ (where $\log$ is the $p$-adic logarithm). This can be obtained from the description of the inverse of Frobenius given in sections 2.9-2.10 of [1]. You can fin …

For $n\geq 1$ and $m\geq 0$, an application of integration by parts ($u=\log^n(1-x)$, $dv=\log^m(x)\,dx/x$) followed by the substitution $x\mapsto 1-x$ shows that $$ \frac{I_{n,m}}{I_{m+1,n-1}}=\frac{ …

This is not a complete answer, but the following result might useful. Basically it says that if the poles of $f(x)$ are negative integers, then the time to compute $\sum_{k=0}^{p-1}f(k)$ is bounded by …

Periods arise from the comparison between Betti and de Rham cohomology for an algebraic variety. The Period Conjecture, due to Grothendieck, is a transcendence conjecture for periods which says that e …

The usual power series for $\log(1+x)$ determines an injection from $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$ into $\mathbb{Q}_p[[x_1,x_2]]$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1 …

The group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times \mathbb{Z}_p]]$ has the structure of a complete Hopf algebra, where $\mathbb{Z}_p\times \mathbb{Z}_p$ consists of precisely the group-like elements. …

For each $n$, the differences $a_{n+1}-a_n$, $a_{n+2}-a_{n+1}$, and $a_{n+2}-a_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since $$ \frac{a_{n+2}-a_{n+1}}{a_{n+2}- …

As far as I know this quantity does not have a name. There is a mention of a general family of $p$-adic limits, of which $S_{k,p}$ is a special case, on p. 31 of [1]. The quantity $S_{k,p}$ can be ex …

Consider the example $f=x_1^2+x_2^2+x_3^2$, $g=x_1^2$. For any prime $p\equiv 3\mod 4$, we have $f\equiv g\equiv 0\mod p\Leftrightarrow x_1\equiv x_2\equiv x_3\equiv 0\mod p$. So for this example $$ \ …