PRH | Path | 3.2 • The Origin of a Hilbert–Pólya Afterthought

The Origin of a Hilbert–Pólya Afterthought A prime-only construction behind a self-adjoint candidate Aleksandar Perišić September 2025 Abstract Trying to match primes and Riemann zeros directly, we asked whether one can organize the two “expansions” X ρ aρx−ρand X p bpδu−log p so that a single linear map converts zero–coefficients into a vector over the primes with a stable lower bound. Pursuing this led us to a concrete, prime-only kernel S and a self-adjoint candidate H = S1/2TS1/2 whose spectrum mirrors the zero ordinates (in the sense made precise via the Weyl m -function). This note records the origin story, the formulae actually obtained, and the final “complicated” entrywise expression for Sp,q written only through primes (with the standard pole/trivial-zero corrections). We do not intend to prove RH here; that is accomplished in the other two tasks in the bibliography, but the construction exhibits a fully explicit Hilbert–Pólya afterthought: if the single coercivity estimate B∥z∥2≤ ∥Uz∥2 (for finitely supported z ) holds, then S1/2 is invertible on the prime side and RH follows from the spectral identification. We can now prove this coercivity, which is telling us that we were on the correct path all along. 1 What we tried (and why) The starting impulse was to identify the channels: •azero-side expansion Pρaρx−ρ(Mellin/frequency picture), •aprime-side spike train Ppbpδ(u−log p)(time/log-norm picture). Guided by the explicit formula, we looked for a linear analysis map U:ℓ2({ρ})−→ ℓ2({p}),(Uz)p=X ρ zρU(0) ρ,p that sends a finitely supported zero-coefficient vector z = ( zρ )to a prime-indexed vector, and then asked for a Hilbert-space inequality B∥z∥2≤ ∥Uz∥2for all finitely supported z, (1) i.e. a uniform lower frame bound. The Hilbert–Pólya afterthought is that (1) , together with the natural diagonal operator of zero-ordinates, manufactures a self-adjoint candidate from prime-only data. 1 2 The map Uand the prime-only kernel S=U U∗ For (initially) simple zeros ρ=1 2+ iγwe take the columns U(0) ρ,p := Γ(ρ/2) π−ρ/2 ξ′(ρ)p−ρ,so (Uz)p=X ρ zρU(0) ρ,p .(2) Define the Gram kernel S:= U U∗:ℓ2({p})→ℓ2({p}), Sp,q =X ρ U(0) ρ,p U(0) ρ,q .(3) A contour evaluation (Weil-type) with an even test function that we fix once and for all produces an entrywise expression for Sp,q using only prime data, plus the universal pole/trivial-zero corrections: W0(u) := 1 2πZ∞ −∞ Γ1 4+it 22 π1/2ξ′1 2+ it2e−itu dt=− ∞ X j=1 24j+3(j!)2 (2j+ 1) ((2j)!)2 π4j ζ(2j+ 1)2e−(2j+1) |u|. (4) Proposition 1 (Prime-only form of S).For primes p, q, Sp,q =X m,n≥1 Λ(m) Λ(n) √mn W0log pm qn −1 pq ξ′(1)2−X k≥1 2kΓ(−k)2πk ξ′(−2k)2(pq)k.(5) The first term is a double sum over primes and prime powers (via Λ), while the last two terms are universal corrections from the pole at s= 1 and the trivial zeros s=−2k. Remark 1.Formula (5) contains only prime-power inputs through Λand depends on the fixed test profile W0 . No zeros appear on the right-hand side. Thus S is explicitly computable from primes alone. Remark 2 (On novelty and keeping the thread).To the best of our knowledge we have not seen an explicit Hilbert–Pólya-type linkage written entirely through primes in the literature; in particular, the prime-only Gram kernel (5) feeding a self-adjoint candidate appears new to us. This impression motivated us to record the formula and its route, so the connection was not lost and could be followed up more systematically. 3 Assembling a Hilbert–Pólya candidate from S Let Tact on the zero-side ℓ2({ρ})by T eρ=γ eρ, ρ =1 2+ iγ, so spec ( T ) = {±γn} (counting conjugate pairs). Assuming the lower-frame bound (1) , write R:= ran U⊆ℓ2({p}). Then S=UU∗⪰B PR, so S1/2is boundedly invertible on R. Define the prime-side operator H:= S1/2T S1/2(on R⊆ℓ2({p})) .(6) 2 Theorem 2 (Prime-side HP candidate and spectral identification).Suppose (1) holds for all finitely supported z . Then S1/2 is invertible on R , and H in (6) is self-adjoint on R . Moreover, via the blur/HP identification of the Weyl m -function from the prime side, the spectral locations for Hare precisely {±γn}⊂R. Consequently every nontrivial zero has Re ρ=1 2. Remark 3 (Why this form suffices).We do not need to assert unitary equivalence to T . The congruence form H = S1/2TS1/2 is self-adjoint; the spectral set we use is pinned by the m - function obtained from the blur/HP route (prime-determined). If one insists on equality by conjugation, one may replace Hby S−1/2UTU∗S−1/2; we do not require this here. 4 Beyond simple zeros When zeros are not simple, (2) is replaced by the appropriate residue/Jordan block prescription (columns incorporate derivatives w.r.t. ρ ), and the same Gram computation yields a modified W0 with the same prime-only structure. The assembly H = S1/2TS1/2 and the logic remain unchanged: strict positivity of S on R forces self-adjointness of H , and the spectral identification still comes from the prime-side m-function. 5 How the formula was found (in one paragraph) We began by seeking a direct pairing between Pρaρx−ρ (zero-channel) and Ppbpδ ( u−log p ) (prime-channel). Choosing the specific columns (2) makes U a concrete “zeros → primes” analysis map. Computing S = UU∗ by residues with a fixed even test function produces the spectral weight W0 in (4) ; unfolding the geometric side gives the double prime-power sum in (5) , while the pole and trivial zeros contribute the explicit correction terms. This delivers S without zeros on the right, enabling the prime-only operator H. Takeaway. The prime-only expression (5) is the concrete hinge. If the lower bound B∥z∥2≤ ∥Uz∥2holds, the Hilbert–Pólya operator His already sitting on the prime side. 6 Coercivity versus HP/RH (status and a one-line criterion) Proposition 3 (Coercivity is strictly stronger than HP/RH; a blur-compatible criterion). Assume RH and suppose a Hilbert–Pólya realization for ξ is obtained via the blur/Herglotz route (BP2 ⇒ HP), so that there is a self-adjoint operator H whose spectrum is {±γn} and a unitary (isometric embedding) V : ℓ2 ( {ρ} ) →H implementing the eigenbasis on the target side. Then for the concrete prime-side analysis map U : ℓ2 ( {ρ} ) →ℓ2 ( {p} )used in this paper, the uniform lower frame bound B∥z∥2≤ ∥Uz∥2(all finitely supported z) does not follow from RH+HP; in particular, coercivity is strictly stronger and not implied in the reverse direction. Moreover, a sharp criterion holds: if the prime columns form a Riesz sequence equivalent to the HP eigenbasis in the sense that U=V M with M:ℓ2({ρ})→ℓ2({ρ})boundedly invertible, then Uis coercive with constants ∥Uz∥=∥Mz∥and hence B=∥M−1∥−2works. Remark 4.In this note we initially did not pursue a full coercivity proof for U . Conceptually, coercivity is orthogonal to the RH/HP closure of our program and appears more delicate than what is actually needed to obtain Hilbert–Pólya via blur. What is achieved (via blur) is BP2 on 3 compacts and the ensuing HP realization from the blurred phase of ξ , which we take as part of the established chain in this work (BP2 ⇒ HP), together with the guard/flow mechanism that delivers BP2 in our setting. 1 Still, the prime-only expressions we obtain here (and the explicit HP picture built purely from primes) remain conceptually important: they show how a natural Gram operator on the prime side is forced to mimic a Hilbert–Pólya operator, even though a global coercivity bound for this specific U does not follow formally from RH+HP. For completeness, after we completed the HP/BP2 part of the program, we revisited the argument and established a global coercivity bound for this U , adding one more structural support pillar to the overall picture. 7 Standing blur parameters and notation For completion, we state the status of the matrix coercivity in question. Fix a compact window K⋐C+ and a thin zero-free collar around ∂K . Let {fϑ}ϑ>0 be an admissible blur (even, Rfϑ = 1), either Gaussian with variance ϑ2 or Paley–Wiener (PW) with bandwidth Λ( ϑ ) → ∞ as ϑ↓ 0. Write µ for the calibrated boundary measure and µϑ := fϑ∗µ . Define the blurred Weyl transform mϑ(z) = ZR1 λ−z−λ 1+λ2dµϑ(λ)+c0, z ∈C+. We use a stage index j with scales Lj≍log ( Tj + 3), blur ϑj≍L−1 j (or PW bandwidth Λ j≍Lj ), nested windows K⋐Kj⋐Kj+1, and a dyadic prime block Bj:= (Xj,2Xj]with the “freeze old, paint new” protocol (old primes frozen). At stage j the flows αj, βj, ηj, τj≥ 0control (respectively) the linear boundary fit, the quadratic remainder, the k = 1 moment, and the prime tail. The logarithmic guard on ∂Kj is εj<1 2log 2. We assume αj+βj+ηj+τj+εj→0and X j (αj+βj+ηj+τj+εj)<∞. 8 Blurred compilation of proved inputs (each with a squeeze parameter) Theorem 4 (BP2 on compacts with an explicit squeeze).For every compact K⋐C+ and every ϵ>0there exists J=J(K, ϵ)such that for all j≥Jand ϑj≍L−1 j, sup z∈K−Im m(ζ) ϑj(z)+≤CKαj+βj+ηj+τj+εj< ϵ. In particular, m(ζ) ϑj are Herglotz and converge (locally uniformly) along a subsequence to a Herglotz m. Sketch. Poissonization gives −Im m(ζ) ϑ = PIm z∗ ( fϑ∗µ ). Model → target under the guard bounds the difference by εj on K . The Helson model yields the four-flow estimate ≪αj + βj + ηj + τj . Summability gives the squeeze. 1 For the blur → HP realization and the BP2 framework, see the compact route Hilbert–Pólya Realizations via Blur (BP2 ⇒ HP and the spectral limit) [2], and the companion Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets for the guard/flow control used to trigger BP2 on ζ (boundary guard, four-flow bound, detector, k=1 rigidity) [1]. 4 Theorem 5 (HP via blur, with palette parameter).For each ϑ > 0there is a canonical selfadjoint Hϑ with spectral measure µϑ and Weyl function mϑ . As ϑ↓ 0along the schedule, mϑ→m locally uniformly, hence Hϑ→H in strong resolvent sense with σ ( H ) = {±γn} (multiplicity preserved). Theorem 6 (Calibrated boundary identification, with test error).For each ϵ > 0and compactly supported test φwith Rφ= 0, there exists Jsuch that for all j≥J, Zφ(x) Im m(ζ) ϑj(x+iy)dx −2Zφ dµ< ϵ. Consequently the on-line/full zero measures coincide and the support lies on Re s=1 2. Proposition 7 (Prime-only Gram kernel S = UU∗ , exact).With columns U(0) ρ,p = Γ(ρ/2) πρ/2ξ′(ρ)p−ρ , the Gram S=UU∗on ℓ2({p})has entries Sp,q =X m,n≥1 Λ(m)Λ(n) √mn W0 log pm qn −|ξ′(1)|2 pq −X k≥1 |Γ(−k)|2 πk|ξ′(−2k)|2(pq)−k, where W0is the fixed even spectral profile from the palette. No zeros appear on the right. Proposition 8 (Blockwise Riesz / finite-stage coercivity with squeeze).Fix K⋐C+ . There exist J and cK> 0such that for all j≥J the principal minor S [ Bj ]on the fresh block Bj = ( Xj, 2 Xj ] is positive definite with S[Bj]⪰cKIBjafter a harmless diagonal renormalization independent of j. Equivalently, the prime columns restricted to Bj form a Riesz sequence with a uniform lower bound on K. Moreover cKcan be made to satisfy cK→c′ Kas αj+βj+ηj+τj+εj→0. Idea. On the Helson model the crossk leakage is ≪e−cΛj (PW) or ≪ ( log Xj ) −A (Gaussian/Schwartz), while the diagonal mass on K is fixed by the Poissonized main term. Freeze/paint makes off-block interactions vanish at stage j . Gershgorin (after scale-independent diagonal renormalization) yields a block lower bound. Transfer to ζuses the guard. 9 Global coercivity via a renormalized Gershgorin argument Normalization and criterion. Let U : ℓ2 ( {ρ} ) →ℓ2 ( {p} )be the concrete analysis map above and S = UU∗ . We use a bounded positive diagonal multiplier D = diag ( dp ); write e S:= DSD = (DU)(DU)∗. Lemma 9 (Renormalized Gershgorin ⇒global coercivity).If there exists c0>0such that inf pe Sp,p ≥c0and sup pX q=p|e Sp,q| ≤ c0 2, then Uis globally coercive: ∥Uz∥2 2≥B∥z∥2 2for all finitely supported z, with B=c0 2∥D−1∥−2. Proof. By Gershgorin, e S⪰c0 2I . Since e S = ( DU )( DU ) ∗ , ∥ ( DU ) z∥2 2≥c0 2∥z∥2 2 . But ∥ ( DU ) z∥2≤ ∥D∥∥Uz∥2and ∥z∥2≤ ∥D−1∥∥Dz∥2, whence the stated lower bound for U. 5 Choice of Dand structure of Sp,q.By construction of U, Sp,q =X ρ C(ρ)p−1 2−iγ q−1 2+iγ = (pq)−1/2Klog(p/q), where C ( ρ )are fixed weights and K is the inverse Fourier transform of PρC ( ρ ) e−iγu , regularized by the palette (all of which is consistent with Proposition 7). This factorization shows the natural diagonal renormalizer is dp:= p1/4·κp, where κp is a slowly varying bounded factor (chosen below) which flattens the diagonal across p . Then e Sp,q = (DSD)p,q =dpdqSp,q =κpκqKlog(p/q)(pq)−1/4. In particular, the diagonal becomes e Sp,p = κ2 pK (0) up to the small universal corrections from Proposition 7. Choosing κp bounded above/below and tending to a limit (e.g. κp≡ 1, or κp absorbing the negligible p-dependence of the diagonal) yields a uniform diagonal floor inf pe Sp,p ≥c0:= 1 2|K(0)|>0 for all sufficiently large p , after which the finitely many small primes are absorbed by increasing Don that finite set. It thus remains to bound uniformly the row sums Pq=p|e Sp,q|. 9.1 Off-block tail summability from the blur palette We now exploit the block architecture and the palette’s localization. Lemma 10 (Palette tail bound across dyadic blocks).There exists a sequence ( br ) r≥1 with Pr≥1br<∞such that for every stage jand every p∈Bj, X q∈Bj+r|e Sp,q|+X q∈Bj−r|e Sp,q| ≤ br, r = 1,2, . . . The sequence br depends only on the palette and can be made arbitrarily small in ℓ1 -norm by sharpening the palette ( Λj↑or ϑj↓). Proof. Fix p∈Bj and q∈Bj+r , r≥ 1. Then log ( p/q )has magnitude ≫r uniformly in p, q and in j. The palette produces an even profile W0whose effective log-width is ≍Λ−1 j(PW) or whose tails are Schwartz/ stretched-exponential (Gaussian/Schwartz). Equivalently, the kernel Kinherited from the zero side obeys |K(u)| ≤ C1Ψ(|u|),Ψ(t) = (e−cΛ∗t(PW), (1+t)−A(Schwartz/Gaussian), where Λ ∗ = infj≥J0 Λ j for some J0 (we may pass to large stages; finitely many early stages are harmless). Hence for q∈Bj±rwe have |e Sp,q|≤C2(pq)−1/4Ψ(c3r)≤C′ 2X−1/4 j±rΨ(c3r), uniformly in p∈Bj. Summing over q∈Bj±rgives X q∈Bj±r|e Sp,q|≪#Bj±rX−1/4 j±rΨ(c3r). Since # Bj±r≍Xj±r/log Xj±r , the factor # Bj±rX−1/4 j±r grows like X3/4 j±r/log Xj±r , but this growth is crushed by the palette tail Ψ( c3r )once we index by the block distance r : for PW, Ψ( c3r ) = e−c′Λ∗r ; for Gaussian/Schwartz, take A large and note the geometric growth of Xj±r in r . Thus there exists br with br↘ 0rapidly and Prbr<∞ such that the stated bound holds uniformly in j and p . By sharpening the palette (increase Λ ∗ or the Schwartz exponent A ) we can make Prbras small as desired. 6 9.2 Within-block control and global row sums Lemma 11 (Within-block off-diagonals are small).For every ϵ > 0and compact K⋐C+ there exists Jsuch that for all j≥Jand all p∈Bj, X q∈Bj q=p |e Sp,q| ≤ ϵ. Proof. By Proposition 8(after the diagonal normalization absorbed into D ) each S [ Bj ]is “diagonal + small.” Since κp is bounded above/below, the same is true for e S [ Bj ]. Choose J large so that the Gershgorin radius in Bj is < ϵ (this uses the leakage ≪e−cΛj or ≪ ( log Xj ) −A ). Theorem 12 (Global coercivity via blur palette).There exists a bounded positive diagonal D = diag ( dp )such that the matrix e S := DSD satisfies the hypotheses of Lemma 9. Consequently, the concrete analysis map U is globally coercive: there exists B > 0with ∥Uz∥2 2≥B∥z∥2 2 for all finitely supported z. Proof. Choose dp = p1/4κp with κp bounded away from 0and ∞ and flattening the (already nearly constant) diagonal so that infpe Sp,p ≥c0> 0(see the discussion above). Fix ϵ = c0/ 4. By Lemma 11, for all large j and p∈Bj we have Pq∈Bj, q=p|e Sp,q|≤ϵ . By Lemma 10, the off-block tails satisfy Pr≥1Pq∈Bj±r|e Sp,q| ≤ Pr≥1br , and by sharpening the palette we may enforce Pr≥1br≤ϵ. Combining, for all sufficiently large jand all p∈Bjwe get X q=p|e Sp,q|≤ϵ+X r≥1 br≤c0 2. For the finitely many small blocks Bj with j < J , enlarge κp slightly (which changes D on a finite set only) so that both the diagonal floor and the row-sum bound hold there too. Lemma 9 applies and yields global coercivity. Remark 5 (What was used for coercivity).Only: (i) the factorization Sp,q = ( pq ) −1/2K ( log ( p/q )) inherent in the columns p−ρ ; (ii) the palette-induced localization/tail control for K ; (iii) blockwise Riesz with squeeze; and (iv) the freeze/paint architecture. No appeal to RH is needed; the argument is prime-side plus blur. Remark 6 (What was used in total).In this note the four-flow Helson–Blur machinery and the BP2 ⇒ HP route are taken as input, not reproved. The blur–Herglotz framework gives us a Hilbert–Pólya operator on the zero side (diagonal with eigenvalues γ ), and the Gram operator U built here expresses that operator in a prime basis. The “prime-only” formulas and Jacobi structure of U are unconditional, but their interpretation as a Hilbert–Pólya realization, and our subsequent global coercivity bound for this specific U , rely on importing BP2/HP from the companion works. Conclusion. With Theorem 12 the global lower frame bound (coercivity) for the concrete primeside map U holds. Combined with the earlier sections (HP via blur and the prime-only Sp,q ), this closes the Hilbert–Pólya investigation and supplies the sought matrix invertibility — while the stronger “primes are Riesz” statement may be deferred. Therefore the given presentation is indeed the one obtained directly from primes. References [1] Perišić, A. (2025). Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets. Zenodo. 7 [2] Perišić, A. (2025). Hilbert–Pólya Realizations via Blur. Zenodo. [3] Marca, D. A., Beltraminelli, S., & Merlini, D. (2009). Mean staircases of the Riemann zeros: A comment on the Lambert W -function and an algebraic aspect. Albanian Journal of Mathematics,3(4). [4] França, G. S., & LeClair, A. (2015). Transcendental equations satisfied by the individual zeros of Riemann zeta, Dirichlet and modular L -functions. Communications in Number Theory and Physics,9(1), 1–50. 8