p-Adic Numbers, Ultrametric Analysis and Applications

In this paper we establish why the p-adic zeta function has a Dirichlet series expansion. We compute an improved expansion, which allows us to express it as a power-series modulo pⁿ. Using this expansion, we compute all the zeros of L_p(s, χω^j) for those quadratic characters χ of conductor < 200. For the calculation we use a PARI-GP Program.

In the paper we show that given a polynomial f over ℤ = 0, ±1, ±2, ..., deg f ⩾ 2, the sequence x, f(x), f(f(x)) = f⁽²⁾(x), ..., where x is m-adic integer, produces a uniformly distributed set of points in every real unit hypercube under a natural map of the space ℤ_m of m-adic integers onto unit real interval. Namely, let m, s ∈ ℕ = {1, 2, 3, ...}, m > 1, let κ_n have a discrete uniform distribution on the set {0, 1, ..., mⁿ - 1. We prove that with n tending to infinity random vectors

\(\left(\frac{\kappa_n}{m^n}, \frac{f(\kappa_n){\rm{mod}} m^n}{m^n}, \ldots, \frac{f^{(s-1)}(\kappa_n) {\rm{mod}} m^n}{m^n}\right)\)

Let ℚ_p be the field of p-adic numbers, a function f(x) belongs to the the Lebesgue class L^ρ(ℚ_p), 1 ρ ≤ 2, and let \(\hat{f}(\xi)\) be the Fourier transform of f. In this paper we give an answer to the next problem: if the function f belongs to the Dini-Lipschitz class DLip(α, β, ρ; ℚ_p), α > 0, β ∈ ℝ, then for which values of r we can guarantee that \(\hat{f} \in {L^r}(\mathbb{Q}_p)\)? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on ℝ.

It is proved that the Schrödinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Integrals of motion are presented. A similar statement is proved for classical dynamical systems in terms of Koopman’s approach to dynamical systems. Examples of explicit reduction of quantum and classical dynamics to the family of harmonic oscillators by using direct methods of scattering theory and wave operators are given.

We explain how the construction of the real numbers using quasimorphisms can be transformed into a general method to construct the completion of a field with respect to an absolute value.