function based zeta function theory to solve the Riemann Hypothesis and a
proposed two semi-circle method to solve the Goldbach conjecture.
Hypothesis states that all non-trivial zeros of the zeta function have
real-part one-half. The conceptual linkage to the UFT pages is given by the
Hilbert-Polya conjecture and the Berry-Keating conjecture.
to the current Gaussian function based zeta function theory a Kummer function
based zeta function theory is proposed to enable the verification of several RH
Goldbach conjecture states that every positive even number n > 2 is the sum
of two primes.
The Hardy-Littlewood circle method is based on the set of integers, which are the zeros of the orthonormal (Fourier series based) basis functions of
the L(2) Hilbert space on the unit circle. All attempts applying the
Hardy-Littlewood circle method to prove the binary Goldbach conjecture failed
due to the insufficient (purely Weyl sum based) bounds for the minor arcs.
two semi-circle method is proposed, which is built on non-harmonic Fourier series and related Riesz bases on the unit circle. They are built on the imaginary parts
of the zeros of the considered Kummer function and the real, negative zeros of the Digamma
function. Both sequences enjoy similar properties enabling N. Levinson's "gap and density theorems". The theory of non-harmonic Fourier series is enabled by appropriately defined „retarded / condensed“ sequences in the form
b(n):=(3*a(n)+a(n+1))/4 fulfilling the Kadec condition. Those sequences are supposed to govern the "even-semi-circle" resp. the "odd-semi-circle". At the same time, both semi-circles can be mapped onto
the negative real line until +1/2 (hosting all negative zeros of the Digamma function) and the positive critical line (not hosting any zero of the considered Kummer function).