A new tool to prove the Goldbach conjecture

The binary Goldbach Conjecture states that every positive even number n > 2 is the sum of two primes.

The Hardy-Littlewood circle method is based on complex-valued power series defined on the "open unit disk" domain. All attempts applying this method to prove the binary Goldbach conjecture failed due to the insufficient (purely Weyl sum based (!)) bounds for the minor arcs.

The proposed alternative Kummer function based Zeta function theory provides appropriate complex-value zeros of the considered Kummer function. Their only real, positive imaginary parts lie all right to the critical line. (They basically show the same appreciated properties than the real, negative zeros of the Digamma function.) The two sequences build a non-symmetric set of numbers with index n=+/-1, +/-2, … forming two Riesz bases on the space of complex-valued functions on the unit circle. Their combination becomes significant in the context of the closure of sets of complex exponential functions satisfying the Paley-Wiener 1/4-criterion and non-harmonic Fourier series, (LeB), (LeN), (PaR), (YoR).

The Riesz bases system provides a pair of two independent prime number density functions on the unit circle in form of non-harmonic Fourier series. The model is proposed alternatively to the complex-valued power series (only) defined on the "open unit disk" domain of the Hardy-Littlewood circle method. It avoids the tool of Weyl sums for the (Goldbach problem independent) minor arcs error estimates.

Note: The imaginary parts of the zeros of the considered Kummer function lie all on the x-axis right to the critical line, while the real, negative zeros of the Digamma function lie all on the x-axis left to the critical line. For both sequences the N. Levinson's "gap and density theorems" can be applied. Additionally, the "Digamma-zeros" x(n) (the maxima and minima points of the Gamma function) come along with an additional 1/log(n) error convergence property compared with the integers n, (NiN) S. 100.

Note: Let a(n) denote both sequences, the imaginary parts of the zeros of a specific Kummer function and the real, negative zeros of the Digamma function. Let b(n) denote the related „retarded / condensed“ sequences in the form 4*b(n):=3*a(n)+a(n+1). Then, the b(n) fullfill the Kadec condition, i.e., they form two non-harmonic Fourier series for L(2)-Hilbert space functions on the unit circle, (YoR).

Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the binary Goldbach conjecture.pdf

Braun K., The Montgomery-Odlyzko law, eigenvalue spacing in a collection of Gaussian unitary Operators