AN INNER-FUNCTION APPROACH TO THE ZEROS OF THE RIEMANN ZETA FUNCTION IN {ℜS > 1 2 }

Abstract. Starting from the Euler product and the regularized determinant det2(I−A(s)) over primes, we construct an inner function I on {ℜs > 1/2} whose zero set coincides with that of ζ and prove unconditionally that I is a pure Blaschke product (the singular inner factor is trivial). The Riemann Hypothesis is equivalent to the statement that this Blaschke product is empty. We develop three approaches to establishing emptiness. Approach I (CR–Green energy comparison) encounters a Cauchy–Schwarz scaling obstruction. Approach II (Schur/Nevanlinna–Pick certification of the Cayley-transformed arithmetic ratio) avoids Cauchy–Schwarz entirely; a hybrid far-field certificate (interval arithmetic on [0.6,0.7] × [0,20], Pick matrix at σ0 = 0.7, and asymptotic bounds) would, under rigorous verification, yield a zero-free half-plane ℜs ≥ 0.6. Approach III (energy-capacity barrier) supplies an unconditional, effective height bound: for each fixed η ∈ (0,0.1), any zero at depth η from the critical line must have ordinate exceeding an explicit Tsafe(η); extending this to all heights reduces to a single arithmetic packing hypothesis (EFBL). All three pathways, their status, and the open problems are documented in full.