Mitsubishi Digital Electronics Inc. MELSOFT for GX Series – 4th Ed. Developer-1.0. MELSOFT for GX Series – 4th Ed.

Assets in libraries are not transferred to MELSOFT. Traditional developers often provide restricted access to the MELSOFT software user’s system. MELSOFT’s.Q:

Exists the the average of a norm of $p$-adic numbers as $p \to \infty$?

Let $p$ a prime number. Let $x$ a $p$-adic number in $\mathbb Q_p$. Let $\lVert x \rVert = e^{ -v_p(x)}$.
Is there exists a constant $C>0$ such that
$$\lim_{p \to \infty} \dfrac{1}{p} \sum_{n=1}^p \dfrac{\lVert n \rVert}{\lVert n+1 \rVert} = C?$$
Thanks

A:

If I am not mistaken, you may take
$$C = \exp(1),$$
when $p$ goes to infinity, since for any $m \in \mathbb{Z}_p$,
$$\frac{1}{p} \sum_{n=1}^{p} \frac{1}{|m – n|_p} \approx \frac{1}{p} \sum_{n=1}^{p} \frac{\exp(-v_p(m-n))}{\exp(v_p(m-n))} \approx \exp(1).$$
See page 18 in this paper for a proof.

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