# An elementary investigation of the theory of numbers by Peter Barlow

By Peter Barlow

Barlow P. An effortless research of the idea of numbers (Cornell college Library, 1811)(ISBN 1429700467)

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We often call s the spectral parameter. Note that Wl (P, n, s) = Wl (P, n, −s). In the literature one often meets the parametrization λ = s(1−s). In comparing results one then needs to make the translation s → s + 12 . 3. The choice λ = s(1 − s) has the advantage of placing the critical line at Re s = 12 , which reminds us of the Riemann zeta function. 7. 2 Diﬀerential equation. 1 is determined by a function f of one variable: F (gP p(z)k(θ)) if P ∈ X ∞ = e2πinx f (y)eilθ if P ∈ PY . 3. This second order linear diﬀerential operator lP,n has real analytic coeﬃcients, so any function f corresponding to an element of Wl (P, n, s) is a real analytic function on (0, ∞).

Product of the subgroups A = { a(y) : y > 0 } = R>0 and N ∼ vzmn(x)x ∈ R = R. This leaves only the deﬁnition of the multiplication of k(θ) ˜ I invite the and p(z). The deﬁnition above lifts the product of p(z) and k(θ) to G. ˜ and that g → gˆ gives a surjective reader to check that we now have a Lie group G, ˜ → G, with kernel central in G. ˜ homomorphism G ˜ It is generated by ζ = k(π). We denote The group Z˜ is the center of G. ˜ Z˜ ∼ ¯ If we use H as a model of hyperbolic plane the quotient G/ = G/{±Id} by G.

4 Iwasawa coordinates. The isomorphism of analytic varieties G ˜ = N ˜ A˜K. ˜ It leads to the Iwasawa corresponds to the Iwasawa decomposition G ˜ coordinates on G: p(z)k(θ) → (x, y, θ). In these coordinates W E = ∂θ ± = e±2iθ (±2iy∂x + 2y∂y ∓ i∂θ ) ω = −y 2 ∂x2 − y 2 ∂y2 + y∂x ∂θ . 5 Polar coordinates. The polar decomposition G ˜ ˜ coordinates on G K: k(η)a(tu )k(ψ) → (u, η, ψ) with η, ψ ∈ R, η + ψ mod πZ, u ∈ (0, ∞) and tu = 1 + 2u + 2 u2 + u > 1, u= (tu − 1)2 > 0. 4tu 30 Chapter 2 Universal covering group We have W = ∂ψ .