The Solution to "Landau's Problems"
N. L. Aljaddou
Properties of the "Rho" function
Principal operator: ρ(p) [the prime function - delineating the number of integers between arbitrary sequenced primes].
I] Goldbach's Conjecture
1) Extend the definition of prime number to a generalized linear algebra vector quantity.
2) Thus, primes as collinear graphs mutually designate each other in magnitude sequence.
3) This is a continuous integrable character of the rho function; therefore, in a continuously generated arithmetic progression range (even numbers; multiples of "2"), the prime number field is a piecemeal duality representation of conjunct ordering - reflex symmetry - for arbitrary sets.
II] The Twin Prime Conjecture
1) ρ'(p) = 0
2) Set ∫ρ'(p) to 1
3) Thus, a unilateral unitary metric standard deviation from primacy - is intrinsic in the domain.
III] Legendre's Conjecture
1) Consider ρ(p) to be the inverse function of π(x) (the prime-counting function).
2) The value of the composite identity function, π(ρ(p)), necessarily results in the "prime- generating function" (the linear bijection relation, resulting in generating of baseline primacy).
3) (i) The introduction of the Lebesgue integral as metric results in a null standard deviation from the linear metric established in the prime-generating function - for the minimal coordinate graphing extension case [in two dimensions] - therefore the case of squared variables in the prime-counting function yields a derivative of zero; or primes are arbitrarily generated for successive integral variables, n2.
(ii) Corollary: The Near-Square Primes Conjecture
The Twin Prime Conjecture Proof, Legendre's Conjecture Proof
I] Disquisitiones Arithmeticae, Carl Friedrich Gauss. Yale University Press (03/11/1965).
II] Invariant Measures, Janos von Neumann. American Mathematical Society (03/01/1999).
III] Divergent Series, G. H. Hardy. American Mathematical Society; 2nd Edition (04/05/2000).
IV] Review: Gotthold Eisenstein, Mathematische Werke, Weil, André. Bulletin of the American Mathematical Society. Volume 82, Number 5 (1976), 658-663.