4
$\begingroup$

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence:

$$a_n = \frac{1}{p^{n-1}}\sum_{(i,p)=1, i=1}^{p^n}\log_p f(\zeta_{p^n}^i-1).$$

This is a p-adic analogue of the Mahler measure. Does this sequence converge (perhaps under suitable conditions on $f$) and if so what is known about the value to which it converges?

I have seen a paper by Besser-Deninger but they study the sum over roots of unity with order coprime to $p$ while I am interested in the exact opposite case. Perhaps someone has also studied this variant?

Improve this question
$\endgroup$
2
  • $\begingroup$ A power series $f(t) \in Z_p[[t]]^\times$ can be written as $f(t) = b_0 \prod_{k \geq 1} (1-b_k t^k)$ with $b_0 \in Z_p^\times$ and $b_k \in Z_p$. Some back of the envelope calculations suggest that, if $f(t)=(1-b_k t^k)$, then it is possible that your $a_n$ converge to something like $(p-1) \log_p b_k$. $\endgroup$
    – Laurent Berger
    Commented Apr 8, 2020 at 16:36
  • $\begingroup$ Thank you! That is a helpful observation. $\endgroup$
    – Asvin
    Commented Apr 9, 2020 at 12:22

0

Reset to default

You must log in to answer this question.