Theta and Riemann xi function representations from harmonic oscillator eigensolutions

Abstract

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.

New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

“…In particular, there are at least four, albeit related, points of view that could connect the h coefficients of equations (2.9) and (3.12), other series coefficients and the sum S 2 , with the theories of diffusion, Brownian motion and random walk. Underlying much of this connection is the fact that a Jacobi theta function (Coffey 2002), a solution of the heat equation, provides a basis for a (Mellin transform) representation of the Riemann xi function (Riemann 1859;Biane et al 2001;Coffey 2005aCoffey , 2007c). …”

THE RIEMANN HYPOTHESIS IS A CONSEQUENCE OF $\mathcal{CT}$-INVARIANT QUANTUM MECHANICS

“…From the explicit form of Eq. (2.34) depicting the inner product of two arbitrary states, by choosing for example that l = −2 ⇒ k = 1 4 , one concludes that the pseudo-norm The Mellin transform based on the weighted Θ 2j (t) [41] requires once again to extract the zero mode n = 0 contribution of Θ 2j (t) (to regularize the divergent integrals) in order to arrive at …”

“…The polynomial P j (s) has simple zeros on the critical line Re s = 1 2 , obeys the functional relation P j (s) = (−1) j P j (1 − s) and in particular P j (s = 1 2 ) = 0 when j = odd [41]. It is only when j = even that P j (s = 1 2 ) = 0 and when we can implement CT invariance resulting from the relation (2.39) and which is consistent with the results of Eqs.…”

“…when m = 0, the zero mode n = 0 does not contribute to the sum and the Mellin transform of Θ 2j,2m (t), after exploiting the symmetry of the even-degree Hermite polynomials, is[40],[41] …”

Special functions and the Mellin transforms of Laguerre and Hermite functions

“…Our work is related to the development of extended theta function representations of the Riemann zeta function ζ. We have very recently shown how to correspondingly generalize the important Riemann-Siegel integral formula [9]. Bump and Ng [5] and Keating [14] have shown how to generalize Riemann's second proof of the functional equation of ζ(s) by using Mellin and Fourier transforms of Hermite polynomials.…”