A Kummer function based Zeta function theory is proposed, alternatively to the current Gaussian function based theory. Basically, this is a replacement of the Gaussian function by its Hilbert transform, which is equal to the Dawson function. It results into an alternatively defined entire Zeta-function accompanied by the product representations of the functions Gamma(1-s/2), 2a*sin(a*s) and a*tan(a*s) with a:=pi/2, where at s=1 the latter function has the same singularity as the extended meromorph zeta function on the half-plane Re(s)>0.
The corresponding alternative representation of the duality equation is accompanied by
- an alternative contour integral representation of the zeta-function for Re(s)<1, which is consistent with the Dawson function based Mellin integral transform representation of the alternative entire Zeta-function for 0<Re(s)<3/2
- the set of the imaginary parts of the only complex-valued zeros of the Dawson function enjoying similar appreciated properties as the zeros of the Digamma function.
The sum of the inverse Fourier transforms of log(2a(sin(as))) and log(a(tan(as))) in combination with two appropriately defined „Landau“ sequences with indices domains (4n-3) and (4n-1), n>0, provides an alternative number theoretical density function to li(x).
The Fourier coefficients of log(a(tan(ax))) are given by the sequence h(n)/n with h(n):=H(2n)-H(n)/2, where H(n) denote the harmonic numbers. The corresponding Dirichlet series defines a new approximating zeta-function for Re(s)>1.
The alternating indices domains (4n-3) and (4n-1), n>0, enable a newly proposed two-semicircle method, where each semicircle is governed by one of the two related density functions. It replaces the major/minor arcs division concept of the Hardy-Littlewood circle method.
The alternating indices domains (4n-3) and (4n-1), n>0, also permits a revisit of Kummer's "ideal complex number" concept based on a proposed (Euler,Kummer) = (4n-3,4n-1) "pairing" concept in a quaternionic setting.
In summary, the alternative entire Zeta function, which is represented as a Dirichlet series built on the Hilbert transform of the Gaussian function for Re(s)>1, accompanied by a corresponding alternative integral representation of zeta(s) in the critical stripe
- simplifies the verifications of several RH criteria
- enables the Landau approach to prove the Goldbach conjecture
- permits a re-examination of the Kummer conjecture.