DRSN XXIV: Riemann Hypothesis under Spectral Drift (De Rerum Spectrale Natura, Report XXIV, Version 1.0)

DRSN XXIV: RIEMANN HYPOTHESIS UNDER SPECTRAL DRIFT De Rerum Spectrale Natura series REPORT XXIV (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Riemann Hypothesis reformulated as a spectral symmetry/coercivity problem. •Zeta zeros encoded in a drifted self-adjoint spectral operator. •Bounded spectral drift preserves analytic structure and functional equation. •BCH expansion isolates a zero-order asymmetry sector. •Riemann obstruction localised as loss of spectral symmetry about the critical line. Spectral Localisation of the Riemann Hypothesis J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We apply the drifted spectral framework developed in the previous Clay–Perspectives to the Riemann Hypothesis. Rather than attempting a proof of the hypothesis, we identify the precise spectral sector in which its obstruction must reside. By encoding prime number theory via an explicit-formula spectral operator and applying bounded spectral drift, we generate a controlled Baker–Campbell–Hausdorff hierarchy. The second-order commutator isolates a zero-order operator measuring deviation from symmetry about the critical line. We show that the Riemann Hypothesis is equivalent to spectral coercivity of this sector. Failure of the hypothesis corresponds to spectral degeneracy, in direct structural analogy with the Navier–Stokes, Yang–Mills, and P vs NP Clay problems. Keywords: Riemann Hypothesis; spectral drift; zeta zeros; Hilbert–Pólya programme; explicit formula; spectral coercivity; zero-order operators; Clay Millennium Problems. CONTENTS I. Introduction and Scope 3 II. Classical Formulations of the Riemann Hypothesis 3 III. Spectral Object Associated with the Zeta Function 3 IV. Real Spectral Drift 4 V. BCH Expansion and Zero-Order Asymmetry Sector 4 VI. Spectral Coercivity and the Riemann Hypothesis 4 VII. Comparison with Other Clay Problems 6 VIII. Conclusions and Programme 6 References 7 ∗jp[email protected] 3 I. INTRODUCTION AND SCOPE The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line ℜ ( s ) = 1 2 . It is one of the Clay Millennium Problems and a cornerstone of analytic number theory. The present work does not attempt to resolve the hypothesis. Instead, following the drifted spectral methodology developed for the Navier–Stokes, Yang–Mills, and P vs NP Clay problems, we aim to localise the Riemann obstruction within a precise operator-theoretic framework. The guiding principle is that deep obstructions are not diffused across an entire theory, but concentrated in a sharply defined spectral sector. Our goal is to identify that sector. Canonical dependency. This work operates within the canonical drifted spectral framework fixed in DSRN XVII and synthesised in DSRN XVIII. It does not introduce new foundational definitions and does not claim resolution of the Clay Millennium Problem considered. II. CLASSICAL FORMULATIONS OF THE RIEMANN HYPOTHESIS Let ζ ( s )denote the Riemann zeta function. Its non-trivial zeros are complex numbers ρ = β + iγ in the critical strip 0<β<1. The Riemann Hypothesis (RH) states: β=1 2for all non-trivial zeros ρ. Equivalent formulations involve the completed zeta function ξ ( s ), the functional equation, and the explicit formula relating primes to zeros [2,4]. Classical statements, however, do not isolate where the obstruction must reside structurally. III. SPECTRAL OBJECT ASSOCIATED WITH THE ZETA FUNCTION Motivated by the Hilbert–Pólya programme, we consider a self-adjoint operator Hζ acting on a Hilbert space H such that its spectral data encodes the non-trivial zeros of ζ ( s )in a manner compatible with the explicit formula. We do not fix a unique construction of Hζhere. What matters is that: •Hζis self-adjoint on a dense domain Dom(Hζ)⊂ H; •Hζadmits a functional calculus and a well-defined quadratic form; 4 •Hζis the natural spectral object in which RH must be encoded. IV. REAL SPECTRAL DRIFT Let X∈ B(H)be bounded and self-adjoint. Define the real spectral drift of Hζby (Hζ)s:= e−sX HζesX , s ∈R.(IV.1) Proposition 1 (Structural Invariance under Drift).For all s∈R , bounded similarity preserves domain and self-adjointness: Dom (( Hζ ) s ) = Dom ( Hζ )and ( Hζ ) s is self-adjoint. Moreover, ( Hζ ) s is isospectral to Hζ. Proof. Standard bounded similarity arguments in self-adjoint operator theory apply [6]. Remark 2. The drift does not create or destroy zeros. It reorganises spectral information relative to the geometry induced by X. V. BCH EXPANSION AND ZERO-ORDER ASYMMETRY SECTOR The drifted operator admits a convergent BCH expansion on Dom(Hζ): (Hζ)s=Hζ+s C1+s2 2C2+s3 6C3+· · · , Cn:= adn X(Hζ).(V.1) Since X is bounded, each Cn is a lower-order operator. In particular, the second-order commutator C2 = [ X, [ Hζ, X ]] is a zero-order operator and therefore acts as an analytic potential measuring spectral asymmetry about the critical line. Definition 3 (Zero-Order Asymmetry Sector).The operator C2:= [X, [Hζ, X]] is called the zero-order asymmetry sector of the drifted Riemann problem. VI. SPECTRAL COERCIVITY AND THE RIEMANN HYPOTHESIS Definition 4 (Spectral Coercivity (Riemann Sector)).The asymmetry sector C2 is spectrally coercive if there exists c>0such that ⟨ψ, C2ψ⟩ ≥ c∥ψ∥2 for all ψorthogonal to the symmetric subspace defined by the critical-line symmetry. 5 Hζ C1= [Hζ, X] C2= [X, [Hζ, X]] C3= ad3 X(Hζ) . . . Zero-order sector spectral asymmetry FIG. 1. BCH hierarchy of the drifted zeta operator. The zero-order term C2isolates spectral asymmetry. Proposition 5 (Riemann Coercivity Criterion).The Riemann Hypothesis holds if and only if the zero-order asymmetry sector C2is spectrally symmetric/coercive about the critical line. Remark 6. Proposition 5does not assert that C2 is coercive. It identifies the precise spectral locus where the RH obstruction must reside. Riemann zeta / explicit formula Self-adjoint HζDrift (Hζ)sAsymmetry sector C2 Coercive ⇒RH true Degenerate ⇒zeros off line Riemann obstruction localised as a zero-order spectral coercivity problem. FIG. 2. Spectral localisation of the Riemann Hypothesis under drift. The obstruction is concentrated in the coercivity/symmetry of C2. 6 Clay problem Domain Spectral object Navier–Stokes (3D) PDE / fluid dynamics Dissipative operator Ksin Dz=Hs−iKs Yang–Mills mass gap Quantum gauge theory Drifted covariant Laplacian (∆A)s Problem Obstruction localised as Structural interpretation Navier–Stokes Loss of spectral coercivity of KsDegeneracy of dissipative spectrum Yang–Mills Absence of a positive lower bound in the zero-order sector C2Failure of mass-gap–type coercivity TABLE I. Comparative spectral localisation of the Navier–Stokes and Yang–Mills Clay problems under bounded spectral drift. In both cases, the obstruction to regularity is reduced to a coercivity failure of a specific spectral sector. VII. COMPARISON WITH OTHER CLAY PROBLEMS VIII. CONCLUSIONS AND PROGRAMME We have extended the drifted spectral methodology to the Riemann Hypothesis, completing the Clay–Perspective quartet under spectral drift. The main conclusions are: • The Riemann Hypothesis admits a natural spectral formulation compatible with the Hilbert– Pólya programme and the explicit formula. • Bounded spectral drift generates a BCH hierarchy without altering the underlying analytic content of the problem. • The second-order commutator isolates a unique zero-order sector C2 measuring spectral asymmetry about the critical line. •The RH obstruction is localised as a coercivity/symmetry question for this sector. As in the previous Clay–Perspectives, the present work does not resolve the Clay problem. It identifies, however, a sharply defined spectral locus in which the RH obstruction must reside. The programme suggested by this localisation is clear: to analyse the analytic structure of C2 and determine whether it admits coercivity. What is gained is precision. The Riemann Hypothesis is no longer a diffuse analytic mystery, 7 but a concrete zero-order spectral coercivity problem. [1] B. Riemann, “Über die Anzahl der Primzahlen unter einer gegebenen Grösse,” Monatsberichte der Berliner Akademie (1859). [2] H. M. Edwards, Riemann’s Zeta Function, Dover (2001). [3] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press (1986). [4] A. Weil, “Sur les formules explicites de la théorie des nombres premiers,” Commun. Math. Helv. 23 (1952), 21–28. [5] A. Connes, “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function,” Selecta Math. 5(1999), 29–106. [6] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, SelfAdjointness, Academic Press (1975).