PRH | Aux | 4.4 • Li Coefficients from the Helson–Blur

Li Coefficients from the Helson–Blur Derivative Addendum: Blurred, Windowed, Operator, Stability, and Explicit Constants Aleksandar Perišić August 2025 Abstract We add four (routine, but new-to-this-platform) Li refinements to the Helson–blur / BP2 ⇒ Hilbert–Pólya program: (i) Blurred Li coefficients λ(ϑ) n with a quantitative lower bound in terms of the four-flow+guard budget; (ii) a finite-window/guard certification that recovers nonnegativity from local collars up to height T ; (iii) an operator-theoretic Li identity via the HP operator and its Cayley transform, yielding positivity as a PSD quadratic form; (iv) a Carathéodory-type stability for Φ ′ ( z )implying soft lower bounds for weighted Li combinations. We then make all constants explicit, tracking dependence on the blur scale ϑ≍L−1 , the Poisson kernel norms, and the pushforward γ7→ eiθγ . For completeness we keep two short classical proofs of RH⇒λn≥0. 1 Standing assumptions and notation (from the mother papers) Let ξ(s) = 1 2s(s−1)π−s/2Γ s 2ζ(s), F(z) = ξ1 2+ iz, z ∈C, F(−z) = F(z). Nontrivial zeros are ρ = 1 2 + i γ with multiplicity mγ . Along the critical line Θ( t ) = arg ξ ( 1 2 + i t ). The calibrated boundary measure is µ=1 πdΘ −dµ∞. Let m(ζ) ϑdenote the blurred Weyl transform (blur scale ϑ≍L−1). On each compact K⋐C+, sup z∈Kℑm(ζ) ϑ(z)− ℑm(χ) ϑ(z)≤α+β+η+τ+ε=: B,(1.1) as organized via the four-flow budgets and boundary guard in the companion notes [ 9 , 10 ]. The Herglotz identification yields ℑm(x+ iy) = π(Py∗µ∗)(x) + a y, a ≥0, and the calibration implies π µ∗= 2 νon = 2 νfull. Li data. The Li coefficients are λn=1 (n−1)!hdn dsnsn−1log ξ(s)is=1 = ∗ X ρ1−1−1 ρn, n ≥1,(1.2) and the Keiper–Li generating function Φ(z) := log ξ1 1−z=X n≥1 λn nzn,Φ′(z) = X n≥1 λnzn−1,|z|<1. 1 2 RH ⇒Li nonnegativity (classical proofs) Under RH, for ρ=1 2+ iγ, 1−1 ρ=−1 2+ iγ 1 2+ iγ=eiθγ,⇒λn= 2 X γ>0 mγ1−cos(nθγ)≥0. Equivalently, push forward the HP/Herglotz spectral measure on {γ} by γ7→ eiθγ to a positive measure νon T, then λn= 2ZT1−cos(nθ)dν(eiθ)≥0. 3 Refinement I: Blurred Li with quantitative lower bounds Fix y∈ (0 , y0 ]and a compact K intersecting {x + i y : x∈R} ; write IK := {x∈R : x + i y∈K} and LK := |IK| (length). Let fϑ be an even, nonnegative Schwartz approximate identity (Rfϑ= 1). Define the Poisson–blurred boundary profile bϑ,y(x) := (Py∗fϑ∗µ)(x). Push bϑ,y 1IK(x)dx forward by γ7→ eiθγ=−1 2+ iγ 1 2+ iγto a finite positive measure νϑ,y;Kon T. Definition 3.1 (Blurred Li on a window).For n≥1set λ(ϑ,y;K) n:= 2 ZT1−cos(nθ)dνϑ,y;K(eiθ). Theorem 3.2 (Quantitative blurred Li).Along the guard schedule ϑ≍L−1 , there exists an absolute constant C0such that λ(ϑ,y;K) n−λ(≤K) n≤2∥νϑ,y;K−ν(≤K)∥TV ≤2LKsup x∈IKℑm(ζ) ϑ(x+ iy)−bϑ,y(x)≤2LKB, for all n≥1; here ν(≤K)is the pushforward of Px∈IKmxδx. In particular, λ(ϑ,y;K) n≥λ(≤K) n−2LKB. If RH holds (hence λ(≤K) n≥0), the right-hand side gives a scale-quantified lower bound. Proof. 1) By construction ℑm(ζ) ϑ ( x + i y ) = b ϑ,y ( x ) + ∆( x )with | ∆( x ) | ≤ B on IK (from (1.1) and the Poisson/Herglotz identification). 2) The total variation norm contracts under pushforward by any measurable map, hence ∥νϑ,y;K−ν(≤K)∥TV ≤RIK| ∆( x ) | d x≤LKB. 3) Since 0 ≤ 1 −cos ( nθ ) ≤ 2, the integral difference is bounded by 2∥·∥TV. Remark 3.3 (No hidden dependence on n ).All constants are independent of n ; the only window dependence is through LK. 4 Refinement II: Finite-window/guard certification Let T > 0, and let UT be thin collars around small circles enclosing each zero with |γ| ≤ T , disjoint from other zeros and Helson lattices. Assume the logarithmic Rouché guard on UT with level εT and that the four-flow budgets over a rectangle KT = [ −X, X ] × [ y0, Y ] ⊃UT sum to BT(hence supKT|∆|≤BT). 2 Definition 4.1 (Truncated Li sums).λ(≤T) n:= 2 X 0<γ≤T mγ1−cos(nθγ). Proposition 4.2 (Windowed Li positivity with stability).Under the hypotheses above, λ(≤T) n≥0for all n≥1, and the reconstruction of λ(≤T) nfrom collar data is stable: b λ(≤T) n−λ(≤T) n≤C1LKT(BT+εT), for some absolute C1, uniformly in n. Idea. The local guard transfers model-toξ data uniformly on UT ; Poissonization and the γ7→ eiθγpushforward preserve positivity and contract total variation, so estimation reduces to the sup-norm error BTon IKTand the guard level εT. 5 Refinement III: Operator-theoretic Li via the HP operator Let H be the HP operator with spectral measure Pmγδγ for a cyclic vector ψ . Its Cayley transform U = ( H−i 2 )( H + i 2 ) −1 is unitary on the cyclic subspace and has eigenvalues eiθγ with multiplicities mγ. Theorem 5.1 (Li as a PSD quadratic form).For n≥1, λn=(2I−Un−U−n)ψ, ψ =∥(I−Un)ψ∥2≥0. Proof. By spectral calculus: ⟨f ( U ) ψ, ψ⟩ = RTf ( eiθ ) d ν ( eiθ ) , with ν the pushforward of the H-spectral measure. Taking f(eiθ) = 2(1 −cos nθ)gives the Li identity, and 2I−Un−U−n= (I−Un)∗(I−Un). 6 Refinement IV: Carathéodory stability and weighted lower bounds Recall Φ( z ) = log ξ ( 1 1−z )and Φ ′ ( z ) = Pn≥1λnzn−1 . Let Φ ′ ϑ,y;K ( z )be the blurred/Poissonized derivative obtained by replacing νwith νϑ,y;Kin the Herglotz–Carathéodory integral: ℜΦ′ ϑ,y;K(z) = Zπ −π 1− |z|2 |eiθ−z|2(1−cos θ) dνϑ,y;K(eiθ)≥ −C(r)∥νϑ,y;K−ν(≤K)∥TV,|z| ≤ r < 1. Theorem 6.1 (Diskwise lower bound).For each r∈(0,1), ℜΦ′ ϑ,y;K(z)≥ − 2(1 + r) 1−r | {z } =:C(r) · ∥νϑ,y;K−ν(≤K)∥TV ≥ − 2(1 + r) 1−rLKB(|z| ≤ r). Consequently, for any nonnegative trigonometric polynomial p ( θ ) = PN n=1 cn (1 −cos nθ )one has the weighted Li lower bound N X n=1 cnλ(ϑ,y;K) n≥ − 2(1 + r) 1−rLKB · ∥p∥Lip,T, where ∥p∥Lip,Tis an absolute constant multiple of Pncn(Fejér–Riesz). 3 Proof. The disk Poisson kernel satisfies supθ1−|z|2 |eiθ−z|2 = 1+r 1−r for |z| ≤ r . Since 0 ≤ 1 −cos θ≤ 2, the kernel factor contributes at most 2(1 + r ) / (1 −r ). Contraction of total variation under pushforward and Theorem 3.2 give the second inequality. The weighted bound follows by integrating against the positive kernel representing p (Fejér), whose norm is comparable to Pncn. Corollary 6.2 (Cesàro–weighted example).For each r∈(0,1), N X n=1 1−n N+ 1λ(ϑ,y;K) nrn−1≥ − C′(r)LKB, with C′(r)depending only on r(via Poisson and Fejér kernel norms), uniformly in N. 7 Explicit constants and dependencies We collect the concrete bounds used above. V.1. Poisson kernels On R:Py(t) = y π(t2+y2)with ∥Py∥L1(R)= 1,∥Py∥L∞(R)=1 πy ,∥Py∗g∥∞≤1 πy ∥g∥1. On T: for |z|=r < 1, Pr(θ) = 1−r2 |eiθ−r|2,sup θ Pr(θ) = 1 + r 1−r. V.2. Blur kernel Let fϑ be an even Schwartz approximate identity: ∥fϑ∥1 = 1, ∥fϑ∥∞≲ϑ−1 , and fϑ→δ in S′ as ϑ↓ 0. No specific profile is required; all dependence in our bounds is through ∥fϑ∥1 (which is 1). V.3. Pushforward map γ7→ eiθγ Write eiθγ=−1 2+ iγ 1 2+ iγ⇒θγ=π−2 arctan(2γ) (mod 2π), θ′(γ) = −4 1+4γ2. Hence |θ′ ( γ ) | ≤ 4, i.e. the map is globally 4-Lipschitz in angle, and the pushforward contracts total variation: ∥ψ∗µ−ψ∗ν∥TV ≤ ∥µ−ν∥TV (ψ(γ) = eiθγ). V.4. Constants used above For any compact K⋐C+intersecting ℑz=y, writing LK=|IK|, we may take C(y, K)=2LK, C(r) = 2(1 + r) 1−r, C′(r)≍1 + r 1−r (the implicit constants in C′ ( r )depend on the Fejér kernel normalization). All bounds are independent of nand of the precise profile of fϑ. 4 8 Discussion: what’s refined and how it fits • Blurred Li (Thm. 3.2). Ties Li positivity directly to the quantitative four-flow+guard control; constants depend only on window length LK. • Finite-window (Prop. 4.2). A local guard up to height T certifies the positive contribution of those zeros, with explicit numerical stability ≪LKT(BT+εT). • Operator form (Thm. 5.1). Repackages λn as the PSD quadratic form ∥ ( I−Un ) ψ∥2 in the HP model; clean and reusable. • Stability (Thm. 6.1). Diskwise lower bounds for ℜ Φ ′ ϑ,y;K yield soft lower bounds for weighted Li combinations, with explicit C(r) = 2(1 + r)/(1 −r). References [1] X. J. Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325–333. [2] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Oxford Univ. Press, 1986. [3] W. F. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer, 1974. [4] B. Simon, Orthogonal Polynomials on the Real Line, AMS Colloquium Publications, 2005. [5] L. de Branges, Hilbert Spaces of Entire Functions, Prentice–Hall, 1968. [6] P. L. Duren, Theory of HpSpaces, Dover (reprint), 2000. [7] P. Koosis, The Logarithmic Integral I, Cambridge Univ. Press, 1988. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw–Hill, 1987. [9] A. Perišić, Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets, September 2025. Zenodo [10] A. Perišić, Hilbert–Pólya Realizations via Blur, August 2025. Zenodo 5