{167--83. \\Spectral Structure and Zeros of the Symmetrized Difference Function $I(s)$}

Abstract

The central object of this paper is the symmetrized difference function: I(s) = zeta(s)/s - zeta(1-s)/(1-s) initially defined for Re(s) in (0,1). Our approach consists in: Constructing I(s), that is, deriving it from a single foundational integral representation (Riemann's formula via the fractional part), allowing the hyperbolic structures to emerge as an inevitable consequence of algebra and analysis; Transposing to centered coordinates w = s - 1/2 = a + ib in order to exhibit I(a,b) as a hyperbolic Fourier–Laplace transform; Studying the zero spectrum by separating real and imaginary parts, establishing the fundamental system, and proving that the only geometry permitting annihilation is the axis a = 0.