Concerning Riemann Hypothesis

We begin by recalling the all-too-familiar lore that the Riemann hypothesis has been the Holy Grail of mathematics and physics for more than a century [1]. It asserts that all the zeros of ξ(s) have σ = 1 2 , where s = σ ±it n , n = 1, 2, 3 . . . ∞. It is believed all zeros of ξ(s) are simple. The function ζ(s) is related to the Riemann ξ(s) function via the defining relation [1],

so that ξ(s) is an entire function, where

ζ(s) is holomorphic for σ > 1 and can have No zeros for σ > 1. Since 1/Γ(z) is entire, the function Γ s 2 is non-vanishing, it is clear that ξ(s) also has no zeros in σ > 1: the zeros of ξ(s) are confined to the “critical strip” 0 σ 1. Moreover, if ρ is a zero of ξ(s), then so is 1 -ρ and since ξ(s) = ξ(s), one deduces that ρ and 1 -ρ are also zeros. Thus the Riemann zeros are symmetrically arranged about the real axis and also about the “critical line” given by σ = 1 2 . The Riemann Hypothesis, then, asserts that ALL zeros of ξ(s) have Re s = σ = 1 2 . We conclude this introductory, well-known remarks with the assertion that every entire function f (z) of order one and “infinite type” (which guarantees the existence of infinitely many Non-zero zeros can be represented by the Hadamard factorization, to wit [2],

where ’m’ is the multiplicity of the zeros (so that m = 0, for simple zero). Finally, ξ(s) = ξ(1 -s) is indeed an entire function of order one and infinite type and it has No zeros either for σ > 1 or σ < 0.

We invoke the well-known results of Lax-Phillips [3] and Faddeev-Pavlov [4] scattering theory of automorphic functions in the Poincare’ upper-half plane, z = x + iy, y > 0, -∞ < x < ∞, which was motivated by Gelfand’s [5] observation of the analogy between the Eisenstein functions [6] and the scattering matrix, S(λ). Recently, Yoichi Uetake [7] has undertaken a detailed study of Lax-Phillips and Faddeev-Pavlov analysis, by resorting to the technique of Eisenstein transform.

We recast this result by specializing to the case where incoming and outgoing subspaces are necessarily orthogonal, D -⊥ D + and by restricting to the case of Laplace-Beltrami operator with constant ‘x’ and identifying the resulting wave equation, as a one-dimensional Schrodinger equation, at zero energy, with a repulsive, inverse-square potential, V (y) = λy -2 , λ > 0, y > 0 and an infinite barrier at the origin, V (y) = ∞, y 0. We obtain zero-energy Jost [8,9] function F + (k 2 = 0):

where ξ(s) is Riemann’s ξ function [1], Eq.( 1) and we have shifted the variable ’s’ by 1 2 , following the convention of Uetake [7]. Since all zeros of Jost function F + (s) lie on the critical line, Rs = -1 4 , we conclude that the Riemann hypothesis is valid.

As an introduction to set the notation, we begin by presenting the familiar case of the Euclidean plane,

The group G = R 2 acts on itself as translations, and it makes R 2 a homogeneous space. The Euclidean plane is identified by the metric

with zero curvature (K = 0) and the Laplace-Beltrami operator associated with this metric is given by

Clearly, the exponential functions

are eigenfunctions of D:

The upper-half hyperbolic half-plane [3] (called the Poincare’ plane) is identified by

|H| is a Riemann manifold with the metric

It represents a model of non-Euclidean geometry, where the role of non-Euclidean motion is taken by the group G of fractional linear transformations,

with ab

The matrix a b c d and its negative furnish the same transformation.

The Riemannian metric, Eq.( 11) is invariant under this group of motions. The Laplace-Beltrami operator is given by

A discrete subgroup of interest is the modular group consisting of transformations with integer a, b, c, d.

A fundamental domain F for a discrete subgroup Γ is a subdomain of the Poincare’ plane such that every point of Π can be carried into a point of the closure F of F by a transformation in Γ and no point of F is carried into another point of F by such a transformation. F can be regarded as a manifold where those boundary points which can be mapped into each other by a γ in Γ are identified.

Then, a function f defined on Π is called automorphic with respect to a discrete subgroup Γ if

for all γ in Γ. By virtue of Eq.( 15), an automorphic function is completely determined by its values on F. The Laplace-Beltrami operator, Eq.( 14) maps automorphic functions into automorphic functions.

In regular coordinates, z = z + iy, if we require f (z) to be a function of y only, i.e., constant in x, we arrive at

with two independent solutions [6],

where

For s = 1 2 (λ 0 = 1 4 ), the above eigenfunctions become y 1 2 and y 1 2 log y respectively.

We now make an important observation which is crucial, i.e., we can view Eq.( 16) as a Schrodinger equation at zero energy, for an inverse-square potential, i.e.,

where

with the all-important constraint that

In other words, the restriction on variable ‘y’ in the Poincare’ upper-half plane, Eq.( 10) requires that y > 0 and Eq.( 22) ensures that this

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