Weil's quadratic form via the screw function

Abstract: We establish a unified framework for understanding the results on the Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes--Consani (2023), and Connes--Consani--Moscovici (2025+) from the perspective of the screw function introduced in Suzuki (2023). An advantage of the approach via the screw function is that it provides a method to study the Weil quadratic form, which is originally defined in terms of distributions, by means of continuous functions. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function can be obtained as the limit, as $a \to \infty$, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval $[-a,a]$. All these results are obtained without assuming the Riemann Hypothesis. This conjecture may be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the Riemann zeta function.

• The paper develops a unified screw-function framework that identifies the localized Weil operator with the Friedrichs extension of a concrete differential-convolution operator, without assuming RH.
• It proves unconditionally that the lowest eigenvalue λ_a varies continuously with interval size a, establishes its positivity, simplicity, even ground state, and sharp logarithmic asymptotics for sufficiently small a.
• It constructs entire functions W(a,θ;z) whose zeros are all real as spectra of self-adjoint extensions, while conjecturing that their large-a limit yields a reciprocal logarithmic derivative of ξ and could imply RH.

Overview

This paper by Masatoshi Suzuki develops a unified operator-theoretic treatment of Weil's quadratic form $Q_W$, the Hermitian form whose positivity on compactly supported test functions is equivalent to the Riemann Hypothesis (RH). The unifying device is the screw function associated with $\zeta(s)$, introduced in Suzuki's earlier work [(2606.09096) builds on 2307.xxxx / J. Lond. Math. Soc. 2023], a continuous real-valued even function $g$ built from elementary terms, the von Mangoldt sum $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$, and Hurwitz–Lerch zeta corrections. By Kreĭn–Langer theory, $g$ is a genuine screw function (its kernel $g(t-u)-g(t)-g(-u)+g(0)$ is nonnegative) if and only if RH holds. Crucially, all results in the paper are proved without assuming RH; RH enters only in the heuristic section motivating the central conjecture.

The paper re-derives and refines results of Yoshida (1992), Bombieri (2001, 2003), Connes–Consani (2023), and Connes–Consani–Moscovici (2025+) within this single framework. Its principal methodological advantage is that Weil's quadratic form, originally defined via distributions, becomes tractable through continuous kernels.

Explicit realization of the localized Weil operator

For each $a>0$, Connes–Consani–Moscovici established abstractly that the restricted form $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$ is closed, lower bounded, and represented by a self-adjoint operator $A_a$ with discrete spectrum. The paper's first main theorem identifies this operator constructively: with $G_a = P_a G P_a$ the compression to $\zeta(s)$0 of the convolution operator with kernel $\zeta(s)$1, and $\zeta(s)$2 on $\zeta(s)$3 (where $\zeta(s)$4 has Dirichlet conditions), $\zeta(s)$5 is precisely the Friedrichs extension of $\zeta(s)$6. The proof hinges on showing that the exponential basis $\zeta(s)$7 — a core for $\zeta(s)$8 — lies in the form-norm closure of $\zeta(s)$9, via cutoff estimates giving $g$0. This connects the spectral interpretation of $g$1 directly to Yoshida's variational program.

A key technical input throughout is the asymptotic expansion near the origin,

$g$2

with $g$3 even, $g$4, $g$5, and $g$6. From it the paper derives explicit quadratic-form identities, including a Fourier-side formula

$g$7

which reproduces Bombieri's explicit-formula computation exactly, and a distributional kernel representation $g$8 recovering Bombieri's Lagrangian.

Continuity of the lowest eigenvalue

Let $g$9 denote the bottom eigenvalue of $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$0, i.e. the infimum of the Rayleigh quotient $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$1. Continuity of $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$2 in $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$3 had been asserted by Bombieri only under parity restrictions, with incomplete proofs. Here the scaling $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$4 transfers the Rayleigh quotient to the fixed interval $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$5, where the dominant term is an $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$6-independent Dirichlet-type form

$$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$7

whose form norm is equivalent to the logarithmic Sobolev weight $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$8. Compactness of $$\sum_{n\le e^{|t|}}\Lambda(n)/(\sqrt n(|t|-\log n))$$9 yields lower semicontinuity of $g$0 along sequences; upper semicontinuity follows from approximating minimizers by test functions. Hence:

Theorem: $g$1 is continuous in $g$2, without parity assumptions and without assuming RH.

An immediate corollary is a new proof of Yoshida's equivalence between RH and nondegeneracy of $g$3 for every $g$4: since $g$5 for small $g$6 (Yoshida's Lemma 2), failure of RH forces some $g$7 with $g$8, hence degeneracy at intermediate $g$9 by continuity. The paper also verifies the identity $g(t-u)-g(t)-g(-u)+g(0)$0 over even/odd subspaces, clarifying the status of Bombieri's continuity claims.

Small-$g(t-u)-g(t)-g(-u)+g(0)$1 asymptotics and simplicity

Exploiting the same machinery plus Dirichlet form theory, the paper proves that for sufficiently small $g(t-u)-g(t)-g(-u)+g(0)$2, $g(t-u)-g(t)-g(-u)+g(0)$3 is positive and simple, with the sharp asymptotic

$g(t-u)-g(t)-g(-u)+g(0)$4

and that the corresponding eigenfunction is even. Simplicity follows from irreducibility of the Dirichlet form $g(t-u)-g(t)-g(-u)+g(0)$5 (the jumping kernel $g(t-u)-g(t)-g(-u)+g(0)$6 is strictly positive) together with its Markov property, which makes the semigroup positivity improving; two a.e.-positive eigenfunctions cannot be orthogonal. This supplies, for small $g(t-u)-g(t)-g(-u)+g(0)$7, exactly the hypotheses (simplicity and evenness) that Connes–Consani–Moscovici must assume for their reality-of-zeros result at finite-dimensional level.

An unconditional entire function with real zeros

The second half of the paper constructs, in the Hilbert space $g(t-u)-g(t)-g(-u)+g(0)$8 obtained by completing $g(t-u)-g(t)-g(-u)+g(0)$9 under the norm $a>0$0 ($a>0$1), the minimal symmetric operator $a>0$2, shown to have deficiency indices $a>0$3. Its self-adjoint extensions $a>0$4 are parametrized by $a>0$5, and their spectra are the zeros of the entire function

$a>0$6

where $a>0$7 span the deficiency spaces. Since these are spectra of self-adjoint operators, all zeros of $a>0$8 are real, unconditionally. This parallels [Theorem 5.10, CCM25], where reality of zeros of $a>0$9 requires simplicity of $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$0 and evenness of $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$1; here no arithmetic input beyond finiteness of the prime contribution for fixed $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$2 is used, and the construction works directly on $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$3 rather than through finite-dimensional restrictions or zeta-regularized determinants.

A conjectural route to RH

Under RH one may take $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$4, and the global Hilbert space $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$5 (the completion of $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$6 modulo the null space of $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$7) is isometric, via $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$8, to a de Branges space $Q_W^{\,a}=Q_W|_{L^2(-a,a)}$9 on which multiplication $A_a$0 has self-adjoint extension $A_a$1 with spectrum equal to the zero set $A_a$2 of $A_a$3. The translation group $A_a$4 on $A_a$5 is strongly continuous, and Stone's theorem identifies its infinitesimal generator with $A_a$6 — a Hilbert–Pólya operator realized as a generator of translations. Since $A_a$7 acts as $A_a$8 on $A_a$9 and embeds into $G_a = P_a G P_a$0, the extensions $G_a = P_a G P_a$1 are expected to converge strongly in resolvent to $G_a = P_a G P_a$2 as $G_a = P_a G P_a$3.

This motivates the paper's central conjecture: if there exist $G_a = P_a G P_a$4 and a correction $G_a = P_a G P_a$5 such that

$G_a = P_a G P_a$6

uniformly on compacts, then RH holds. The right-hand side reflects the multiplicity-one structure of the extension spectrum. Notably, Section 8 reformulates everything in terms of continuous kernels: the eigenfunctions $G_a = P_a G P_a$7 solve Fredholm equations of the first kind for $G_a = P_a G P_a$8, and using $G_a = P_a G P_a$9 the conjecture becomes a purely analytic limit statement involving ordinary functions. The paper also recasts Bombieri's Problem 1 as the eigenvalue problem for the compact operator $\zeta(s)$00 on $\zeta(s)$01, and shows the Rayleigh quotient equals a generalized eigenvalue problem $\zeta(s)$02 with $\zeta(s)$03, whose spectrum coincides with that of $\zeta(s)$04.

Limitations and open questions

Several points remain conditional or unresolved. The limit formula in Corollary 6 is conjectural: its proof would require controlling the parameter $\zeta(s)$05 in the definition of $\zeta(s)$06, which in turn demands detailed estimates of the prime-term contribution to $\zeta(s)$07 beyond what the unconditional arguments use. It is plausible that the normalization factor $\zeta(s)$08 is unnecessary ($\zeta(s)$09), but this is not pursued. The identification of the kernel of $\zeta(s)$10 with $\zeta(s)$11 — equivalently, of $\zeta(s)$12 with the screw-function kernel — is established only under RH; proving it unconditionally would itself imply RH, but no proof is given. Whether the general self-adjoint extensions $\zeta(s)$13 admit intrinsic characterizations as extensions of $\zeta(s)$14 is left open, as is the relation between $\zeta(s)$15 and the domain of $\zeta(s)$16. Finally, the claimed continuity in Bombieri's Theorem 5 under parity decomposition is verified here only indirectly through the identity $\zeta(s)$17, not by an independent proof of continuity of $\zeta(s)$18.

Conclusion

The paper demonstrates that the screw function attached to $\zeta(s)$19 organizes the known variational and spectral theory of Weil's quadratic form into a coherent framework: it yields an explicit Friedrichs-extension description of the localized Weil operator $\zeta(s)$20, an unconditional proof of continuity of its ground-state energy, small-$\zeta(s)$21 simplicity and asymptotics, and an unconditionally real-zeroed entire function $\zeta(s)$22 whose conjectural large-$\zeta(s)$23 limit is the reciprocal logarithmic derivative of $\zeta(s)$24. The reduction of the remaining gap to RH — establishing the uniform limit formula for $\zeta(s)$25 — is thereby converted into a concrete analytic problem concerning Fredholm integral equations with the continuous kernel $\zeta(s)$26.