www.riemann-hypothesis.de
July 30, 2022 6.
The non-trivial zeros of the Riemann Zeta function arise from inquiries into the distribution of prime numbers. The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?“
The not vanishing constant Fourier term of the Gaussian function in the Poisson summation Formula (which is equivalent to the functional equation of the entire Zeta function) is the root of evil to build a self-adjoint operator with transform Zeta(s), (EdH) 10.3. At the same time the spectrum of a hermitian operator, whose inverse operator is not compact, is not purely discrete. 2. The Hardy-Littlewood circle methodApplying the Hardy-Littlewood circle method to prove the binary Goldbach conjecture failed due to insufficient (purely Weyl sum based, w/o any data from the problem under study) bounds for the minor arcs. The Schnirelmann density of the odd integers is ½, while the S-density of the even integers is zero. However, conceptually the Hardy-Littlewood circle method does not distinguish between odd and even integer.By appealing to a heuristic form of the circle method Patterson‘s heuristic fell short of a proof of his conjecture explaining the suspected bias of the Kummer conjecture (DuA). This was also due to insufficent bounds for the minor arcs. We note that the Patterson conjecture is confirmed conditionally on the assumption of the Generalized Riemann Hypothesis, (DuA). We further note that there is also are refinement from the Patterson conjecture that features an error term capturing square root cancellation, (DuA). We also note that that the cubic large sieve cannot improved relying on the GRH based on a dispersion estimate for cubic Gauss sums, (DuA).
The considered Kummer function are the solution of an underlying self-adjoint Whittaker partial differential equation. The discrete spectrum in case of the Hardy-Littlewood circle method is the set of integers, which are the zeros of the orthonormal basis functions (1,sin(n*), cos(n*)). The imaginary parts a(n) of the zeros of the considered Kummer function enjoys similar appropriate properties than the Zeros of the Digamma function. This property enables the definitions of corresponding Hilbert scales and „retarded/condensed“sequences b(n):=(3*a(n)+a(n+1))/4 with „density“ ½ fulfilling the Kadec condition, (YoR) p. 36. We mention the related Kummer "distribution" conjecture of the cubic exponential sums with p = 1 mod 3. The definition of a Riesz basis appears by weakening the „unitary“ condition of U to bounded bijective operators, ChO)3.6. It is the central concept in the theory of non-harmonicFourier series accompanied with the concept of a Riesz basis. Therefore, the mapping (1, sin(n*),cos(n*)) to (sin(b(n)*),cos((b(n)*) enables a transfer from harmonic Fourier series analysis to non-harmonic Fourier series analysis, whereby the underlying indices domain has Schnirelmann density 1/2.
scope of application: pp. 2-4 Supporting papers
April 18, 2021
July 31, 2019
March 27, 2017 Further supporting data
| ||||||||||||||||||||||||||||||||||||||||||||||||