An infinite set of hybrid functions with one unique member whose verifiable zeros are to be found only on Riemann's Critical Line and nowhere else in the Critical Strip (PDF)

Academic Paper from the year 2022 in the subject Mathematics - Analysis, grade: 2.0, , language: English, abstract: This paper devises a hybrid function, denoted by H_a, (where a, is a real constant), which consists of the linear combination of a novel form of the Riemann zeta function and the abscissa of any point in the complex plane. These functions comprise an infinite set, for the value and algebraic sign of the constant are unconstrained.

Amongst these functions, H_(1/2) is unique, in that, the magnitude of its value at the intersection of any Dirichlet line [1] with Riemann's Critical Line [2] is shown to be absolutely zero and that there are no other zeros of this function anywhere else in the Critical Strip.

There may be other zeros of this function elsewhere in the complex plane, but this paper argues that this can never be proved; this is a feature of any other of the H_a whose zeros can be posited to exist at the intersection of a vertical line passing through any abscissa of choice with a Dirichlet line but, can never be shown to be exactly zero, since this would require that the Dirichlet alternating eta series associated with the real part of these H_a be summed to infinity.

It follows from the above that, for the function, H_(1/2) Riemann's hypothesis is verified.