The Kummer & Dawson functions and the Hilbert space frameworks H(-a) with a=1, 1/2, 0, -1/2, -1 are proposed as alternative frameworks to solve the Goldbach conjecture. The failure of the circle methods to prove both, the 3-primes and the 2-primes problem, is due to not sufficient convergence behavior of minor arcs estimates, which are basically Weyl exponential sums estimates (WaA). We emphasis that the convergence behavior of the minor arcs estimates (for both, the 3-primes and the 2-primes problem) are depending from the exponential sums estimates, only, without taking any further information into account related to the 2-, 3-primes problems. The alternatively proposed hypergeometric series provide improved convergence behaviors.

The mathematical alternative framework in a nutshell

The circle method is based on a generating power series with respect to the prime number series. This power series is related to the homogenous Dirichlet boundary value problem on the unit circle: the potential function solution u of the Dirichlet problem can be representated as generalized Fourier series representation with Fourier coefficients derived from the given boundary value function g. The Poisson integral representation (which can be derived from the Cauchy integral formula which plays a key role defining "arcs" in the circle method) relates (in an appropriately defined Hilbert scale domain) to a self-adjoint singular integral operator with apropriate domain H(-1/2), e.g. (BrK). This means that there is a convergent generalized Fourier series representation of the potential function u on the unit boundary which is proposed as alternative generalized generating series to the circle method power series. The prize to be paid by this alternative generating power and Fourier series representation for a "leveraged" circle method is an analysis in a Hilbert scale framework instead of dealing with e.g. continuous functions.

As a consequence the (Goldbach) problem independent not sufficient convergence behavior of the minor arcs (on the unit circle) is improved in the less regular fractional Hilbert space framework, becoming now a truly "minor" convergence estimate contribution to the (Goldbach conjecture) dependent major arcs estimates. As a consequence the existing sufficient convergence behavior of the major arcs to prove the Goldbach conjecture determines the overall convergence behavior (minor arcs plus major arcs estimates) which then finally proves the Goldbach conjecture.


Here we are:

    

Braun K.,A distributional way to prove the Goldbach conjecture leveraging the circle method