HBP | Inter | 5.1 • Riemann Hypothesis and the Multiplicative Choice

Riemann Hypothesis and the Multiplicative Choice A clear structural link Aleksandar Perišić September 2025 Abstract We present a compact, fully spelled–out conditional route from a specific, explicit choice of multiplication (as formalized in Perišić, Multiplicative Closure of the World, 2025) to the conclusion that all nontrivial zeros of ζ lie on the critical line. The link passes through three items that are part and parcel of that choice: (i) power–compatibility on the discrete side; (ii) the unitary Mellin transform on the midline ℜs = 1 2 ; and (iii) scale–neutral “blur” (Fejér/Paley–Wiener) on the log–line. The only additional hypothesis is a boundary positivity that is exactly what those multiplicative constraints naturally allow (Fejér–Mellin positivity). We do not claim a stand–alone proof of RH here. Rather, we show that, under these axioms, RH (in the sense of location of zeros) follows formally and transparently; we also list the assumptions explicitly, explain why they are intrinsic to the multiplicative choice, and why they do not undermine the structural conclusion. We then explain in two voices—formal and plain–language— why simplicity of zeros is not forced by the basic multiplicative axioms alone, and how a natural “no hidden degeneracy” addendum (dilational irreducibility) aligns simplicity with the same choice. Finally, we explain how the Helson framework (guards, detector, four flows) is used bottom–to–top to derive the needed positivity and to address simplicity, should one seek unconditional rigor beyond this illustrative note. 1 How we fix multiplication We make precise the two commitments about multiplication highlighted in [ 1 ] and add the unique scale–neutral smoothing they permit. Axiom 1 (Power–compatibility).On the discrete/combinatorial side, repetition defines a power operation that descends to lengths to give [ n ] ⊗ [ m ]=[ nm ]. In short: the tensor meant to encode “do mtimes, then forget the path” is multiplication. Axiom 2 (Unitary Mellin midline).On L2((0,∞), dx/x), the Mellin transform Mf(t) := 1 √2πZ∞ 0 f(x)x−1/2−it dx x is unitary, and the involution x7→ 1 /x implements s7→ 1 −s with the unique fixed line ℜs = 1 2 . This is the analytic footprint of our multiplicative choice: energy is preserved exactly at the midline. Axiom 3 (Scale–neutral blur).Smoothing compatible with dilations and dx/x must live on the log–line. We use even approximate identities fϑ that commute with dilations: either Gaussians in log x(Mellin multiplier e−ϑ2t2/2) or Paley–Wiener bandlimits (multiplier 1[−Λ,Λ]). 1 2 Calibrated phase and the blurred Cauchy transform Let ξ ( s ) = 1 2s ( s− 1) π−s/2 Γ( s 2 ) ζ ( s )and write s = 1 2 + it . Put Θ( t ) = arg ξ ( 1 2 + it )with Θ(0) = 0. Fix a smooth archimedean reference dµ∞and define the calibrated phase measure µ:= 1 πdΘ−dµ∞(tempered distribution on R).(1) For even blur fϑ(Axiom 3) write µϑ=fϑ∗µand define, for z=x+iy ∈C+, mϑ(z) := ZR1 λ−z−λ 1+λ2dµϑ(λ)+c0, c0∈Rchosen by ℜmϑ(i)=0.(2) Then ℑmϑ(x+iy)=πZR 1 π y (x−λ)2+y2dµϑ(λ)=π(Py∗µϑ)(x),(3) i.e. ℑmϑis the Poisson extension of the blurred boundary datum. 3 The single extra assumption (intrinsic to the choice) Our multiplicative choice constrains which tests are admissible on the boundary: exactly the Fejér cone on the log–line (even convolution squares). We encode this as positivity for the blurred boundary measure. Axiom 4 (Fejér–Mellin positivity (FP)).For every admissible blur fϑ , the measure µϑ = fϑ∗µ is a nonnegative Borel measure on R. Equivalently (distributionally): for every even Schwartz g , with ϕ = g∗˜g (Fejér square), ⟨µ, ϕ⟩ ≥ 0. 4 A short conditional chain from the choice to the line We now record (with proofs) the compact chain that ties the multiplicative choice to the critical line. Everything below is standard once Axioms 2–3and (FP) are in place. Proposition 1 (BP2 on compacts).Under (FP), ℑmϑ ( z ) ≥ 0for all z∈C+ and all ϑ > 0. In particular, on every compact K⋐C+,infz∈Kℑmϑ(z)≥0(BP2). Proof. Immediate from (3) with µϑ≥0. Proposition 2 (Herglotz limit).Let ϑj↓ 0. A subsequence of mϑj converges locally uniformly on C+to a Herglotz function mwith Nevanlinna representation m(z) = ZR1 λ−z−λ 1+λ2dµ(λ)+c0, where µis the weak-∗limit of µϑjand is positive. Proof. Normal family + Fatou for Herglotz functions; positivity passes to the limit. Lemma 1 (Off-line pairs force a negative bump).Let ρ = β + iγ be a zero of ξ with β = 1 2 , γ > 0. Along the line s=1 2+it, 1 π d dt arg(s−ρ) = 1 2−β π(t−γ)2+ (1 2−β)2,1 π d dt arg s−(1 −ρ)=− 1 2−β π(t+γ)2+ (1 2−β)2. Hence the quartet {ρ, ¯ρ, 1 −ρ, 1 −¯ρ} contributes an odd density whose negative part is localized near t≈ −γ. 2 Proof. d dt arg ( s−ρ ) = ℑd dt log ( s−ρ ) = ℜ1 s−ρ , with s−ρ = ( 1 2−β ) + i ( t−γ ). The partner 1−ρflips the sign. Conjugates reflect t7→ −t. Conditional Theorem 1 (Conditional link: choice ⇒ critical line).Assume Axioms 1,2, 3and 4. Then the limiting calibrated boundary measure µ in Proposition 2is positive. By Lemma 1, any off-line zero would force a negative test against a small nonnegative bump near −γ, a contradiction. Hence all nontrivial zeros of ζlie on ℜs=1 2. Remark 1 (What the theorem is and is not).Theorem 1is a conditional location result: it shows that RH is a direct formal consequence of our explicit multiplicative axioms plus (FP). It does not claim those axioms are already derived within this note; see §5and §7. 5 Assumptions laid bare, and why they fit multiplication What is assumed? Only Axioms 1–3and 4. The first two are the analytic face of the multiplicative choice in [ 1 ]. Axiom 3is the unique scale–neutral smoothing compatible with that choice. The remaining hypothesis (FP) says: once we restrict to Fejér tests that commute with dilations and are even on the log–line, the calibrated boundary datum evaluates nonnegatively. Why (FP) is natural here. The choice of multiplication leaves us with no other invariant cone of tests on the log–boundary: the even Fejér cone is the minimal positivity one can ask for that respects dilations and the midline unitarity. In that sense, (FP) is not a foreign analytic add–on; it is the precise positivity that our choice permits. Does assuming (FP) undermine the conclusion? No: it makes the precise, checkable statement of §5the only bridge one needs to cross. Moreover, in §7we explain how the Helson package is designed to derive (FP)/BP2 from first principles, without assuming it. 6 Simplicity: formal route and plain-language route The chain above forces location but not multiplicity: a positive pure–point measure may place mass mπ at an ordinate, corresponding to a zero of multiplicity m . This is not ruled out by Axioms 1–3alone; the multiplicative choice fixes where the spectrum lives, not yet how many eigenvectors sit at each ordinate. A multiplicative addendum suggesting simplicity (formal). We introduce a natural strengthening tied to multiplication: Axiom 5 (Dilational irreducibility (DI)).The boundary representation of the dilation group on the calibrated datum is cyclic: there exists a single even test (a log–Gaussian, say) whose dilational orbit is total (dense) in the boundary Hilbert Hardy space H2 determined by the choice. Heuristically, (DI) says: there are no nontrivial dilation–stable subspaces on the boundary (no hidden duplicate layers of the same structure). In the Herglotz/Nevanlinna dictionary this forbids spectral multiplicity >1. Proposition 3 (Simplicity from (DI)).Under Axioms 1–3,4, and 5, the representing measure µ in Proposition 2has multiplicity one almost everywhere; in the pure–point situation this means each atom has unit mass. In particular, all critical zeros are simple. Sketch. Cyclicity of the boundary representation corresponds (via the Poisson/Cauchy transform) to cyclicity of the associated model operator, which forces scalar spectral measure (multiplicity one). This is the standard boundary–to–model correspondence in de Branges/Kre˘ın theory. 3 Plain-language takeaway (why simplicity is the default here). • We invented multiplication in the most neutral way: repeat → multiply, no absorbing zero, no preferred unit, no extra knobs. That gives dilation symmetry and the unitary midline. • In that neutral world, you probe the boundary with one even “mother” test (e.g. a log-Gaussian) and all its dilates. If these already generate everything (DI), there is no second hidden layer to support a “double direction” at the same frequency. • Adouble zero would require extra structure you have to explicitly build: a second independent boundary direction (non-cyclicity), a matrix-valued model, or a palette with spectral holes. Those are non-neutral additions. • Because we did not make such extra choices when we picked multiplication, the minimal, scale-neutral model defaults to simplicity. If someone wants double zeros, they must say where and why; absent that, the neutral multiplicative setup has simple zeros. 7 Bottom to top: how Helson makes the link unconditional The Helson framework (in the accompanying notes) was built exactly to derive the positivity needed above and to settle simplicity without positing (DI): • Boundary guards + four flows = ⇒ interior lower bound (BP2) for a protected Helson proxy; transfer brings BP2 back to ζ , justifying the step used in Proposition 2without assuming (FP). • Gaussian detector + k = 1 moment rigidity = ⇒ no off–line mass under a guard and collapse of the proxy to the trivial twist, which upgrades location to simplicity and produces a canonical model. Thus, even if one declines to assume (FP)/(DI), the bottom–to–top Helson path is built to supply them. While we cannot be certain from this single article alone, one practical route is open: if RH is true, then the choice of multiplication is essentially the one articulated here—fixed by replication, scale neutrality, and midline unitarity. It is, in a way, a bit scary (and also clarifying) to note how far mathematics developed without holding this torch in plain sight. 8 Scope, alternatives, and what this note is (and isn’t) This note clarifies: given the precise analytic face of your multiplicative choice, RH is structurally tied to that choice via a single, natural positivity. It is not asserted that RH can only be obtained this way: if some independent proof of RH exists that does not invoke these axioms, then our statement becomes one of compatibility: the multiplicative axioms are a regime where RH is expected and explained. For unconditional completeness and for simplicity/canonicity, we refer to the Helson stack. Bottom line. The choice of multiplication (Axioms 1,2,3) leads directly to RH conditional on the Fejér–Mellin positivity that this very choice singles out. Adding a natural dilational 4 irreducibility (Axiom 5) aligns simplicity with the same choice. When one prefers to avoid any such axioms, the Helson machinery is the vehicle to supply them. References [1] A. Perišić, Multiplicative Closure of the World, Zenodo (2025). [2] A. Perišić, Hilbert–Pólya Realizations via Blur, Zenodo (2025). [3] A. Perišić, Boundary Guards, Detectors, and Four–Flow Budgets, Zenodo (2025). [4] W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer (1974). [Herglotz/Nevanlinna.] [5] L. de Branges, Hilbert Spaces of Entire Functions, Prentice–Hall (1968). [6] E. C. Titchmarsh (rev. D. R. Heath–Brown), The Theory of the Riemann Zeta–Function, 2nd ed., Oxford Univ. Press (1986). 5