Is There a $p$-Adic $\pi$? Haar-Measure Circle Constants over $\mathbb{Q}_p$ and an Adelic Product Formula

In the paper's words

The real number π ≈ 3.14159 is defined as the ratio of a circle's circumference to its diameter. Since Ostrowski's theorem embeds Q diagonally into every completion R and Q_p, one may ask whether this ratio admits a well-defined analogue π_p at each p-adic place, as a step toward testing the conjecture, recorded in the source corpus of this work, that π is the archimedean projection of a deeper adelic object. We work with the Haar measure μ on Q_p² normalized by μ(B₁) = 1, where B₁ is the unit ball of the sup norm.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The real number $\pi \approx 3.14159$ is defined as the ratio of a circle's circumference to its diameter.

   from Abstract

2. The Euler product over all places gives $\prod_p \kappa_p = 6/\pi^2 \approx 0.607927$, and the all-place product including the archimedean factor is $6/\pi \approx 1.909859 \neq 1$.

   from Abstract

3. In mathematical physics, entry [3] (arXiv:math-ph/0512018v2), "On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree," studies phase transitions of a $p$-adic Potts model on a Cayley tree; the title indicates that $p$-adic statistical mechanics supports genuine critical phenomena whose structure depends on the model's parameters, illustrating that $p$-adic models behave in parameter- and place-dependent ways, the phenomenon we quantify for the circle ratio.

   from 2. Background and Related Work

4. Multiplying the archimedean ratio by the product over all finite places, $$\kappa_\infty \cdot \prod_p \kappa_p = \pi \cdot \frac{6}{\pi^2} = \frac{6}{\pi}.$$ Numerically, $6/\pi \approx 6/3.141593 \approx 1.909859$.

   from 4. Analysis

5. The Euler product over all finite places equals $6/\pi^2 \approx 0.607927$, and the all-place product including the archimedean factor is $6/\pi \approx 1.909859 \neq 1$.

   from 7. Conclusion

How it is done

1. Since Ostrowski's theorem embeds $\mathbb{Q}$ diagonally into every completion $\mathbb{R}$ and $\mathbb{Q}_p$, one may ask whether this ratio admits a well-defined analogue $\pi_p$ at each $p$-adic place, as a step toward testing the conjecture, recorded in the source corpus of this work, that $\pi$ is the archimedean projection of a deeper adelic object.

   from Abstract

2. Ostrowski's theorem asserts that $\mathbb{Q}$ embeds diagonally into all of its completions, the archimedean $\mathbb{R}$ and the non-archimedean $\mathbb{Q}_p$ for every prime $p$.

   from 1. Introduction

3. Entry [1], "On the p-adic Beilinson conjecture for number fields" (arXiv:0707.3682v2), concerns a $p$-adic counterpart of the Beilinson conjecture, which in its classical form relates special values of $L$-functions to regulators; its supplied abstract formulates a $p$-adic analogue of Borel's theorem using syntomic regulators and $p$-adic $L$-functions; this is direct evidence that the translation of archimedean special-value statements into $p$-adic language is an active program, the kind of program within which an adelic $\pi$ would have to live.

   from 2. Background and Related Work

4. On $\mathbb{Q}_p^2$ we use the sup norm $$\|x\| = \max\bigl(|x_1|_p,\ |x_2|_p\bigr), \qquad x = (x_1, x_2) \in \mathbb{Q}_p^2,$$ which is the standard ultrametric norm on $\mathbb{Q}_p^2$.

   from 3. Methods

What would falsify it

1. What would falsify or revise these claims.

   from 6. Discussion

Where the claim stops

1. In the Haar-measure formulation, the conjecture fails in its strong form: there is no place-wise deformation $\{\pi_p\}$ interpolating toward $\pi$.

   from 6. Discussion

2. We conclude that the adelic-$\pi$ conjecture, in its Haar-measure formulation, fails place-wise while surviving globally: $\pi$ is a product invariant of the finite places, not a deformation target of place-wise constants.

   from 7. Conclusion

3. We conclude that a place-wise family of circle constants exists but is not a deformation of $\pi$; the adelic-$\pi$ conjecture fails in this Haar-measure formulation.

   from Abstract

Where to read

1. Abstract207 words · 1 min
2. 1. Introduction412 words · 2 min
3. 2. Background and Related Work691 words · 3 min
4. 3. Methods408 words · 2 min
5. 4. Analysis674 words · 3 min
6. 5. Results185 words · 1 min
7. 6. Discussion361 words · 2 min
8. 7. Conclusion118 words · 1 min