bibkey: bombieri2000quadratic
authors: Enrico Bombieri
year: 2000
title: Remarks on Weil's quadratic functional in the theory of prime numbers, I
doi: null
url: https://www.bdim.eu/item?id=RLIN_2000_9_11_3_183_0
claim: Finite spectral approximations have one negative eigenvalue per distinct nonreal conjugate pair. Passage to the complete form retains a zero-limit and linear-dependence alternative; finite inertia is not an all-support positivity certificate.
strata_touched: []
license: citation-only
triage: anchor

Finite inertia and the complete-form sign obligation

Primary source and normalization

Bombieri, Remarks on Weil’s quadratic functional in the theory of prime numbers, I, Rendiconti Lincei, Matematica e Applicazioni, series 9, 11 (2000), no. 3, 183–233. The BDIM record links the original PDF. The statements and surrounding arguments cited below were inspected in that scan. This is a source comparison, not an independent audit of its full proof, a new inertia theorem, or Lean certification.

The source writes $\rho =1/2+i\gamma$. Thus its nonreal $\gamma$ means an off-critical zero; its $\gamma$ is not merely the real ordinate of $\rho$. Section 8, printed p.208, starts with a finite multiset $\Gamma$ invariant under complex conjugation and multiplication by $-1$, with positive integral multiplicities preserved. The source’s matrices and norms are retained; no identity with the project’s complete physical operator is assumed.

Reuse the finite results with their finite quantifier

Theorem 8, printed p.213, gives one negative eigenvalue of $\mathcal{H}(\Gamma ;t)$ per distinct nonreal conjugate pair in $\Gamma$. Theorem 9, printed p.217, gives the corresponding count for ${\mathcal{K}}_E(\Gamma )$, where $E$ is a finite union of bounded closed intervals, and separate even and odd counts when $E=-E$.

The paragraph preceding Theorem 8, printed p.213, and Lemma 12, printed p.216, keep multiplicity separate from this count. Repeating the same spectral node adds null copy directions; the nonzero spectrum is obtained from the matrix on distinct nodes with their positive multiplicity weights. A zero’s analytic multiplicity therefore does not create independently interpolable copies of the same scalar evaluation.

These are finite spectral approximation theorems. They do not state that an arbitrary cutoff has the same negative index as the complete Weil form on its test-function domain.

The infinite passage keeps the vanishing-synthesis alternative

Sections 10–11 retain the standing hypotheses on the infinite multiset: $\Gamma =\overline{\Gamma }=-\Gamma$, with symmetry-preserved positive integral multiplicities, and $$\sum _{\gamma \in \Gamma }(1+|\gamma |{)}^{-1-\epsilon }<\infty \text{for every }\epsilon >0,$$ counting multiplicity. They additionally assume finitely many, but at least one, off-critical zeros. Theorem 10, printed p.221, concerns the first eigenvalue problem and states the alternative $T[f\ast f^{\ast }]=\lambda \Vert Df{\Vert }^2<0$ or $f\equiv 0$. It also states that $f\equiv 0$ for every sufficiently small positive $t$, below an explicitly computable $t_0$. This is distinct from Theorem 11, printed pp.223–224, for the $L^2$ eigenvalue problem on a finite union of intervals.

In Theorem 11 the limiting coefficients have unit ${\ell }^2$ norm and at least half their ${\ell }^2$ mass on the off-critical set. Their synthesis $F(u)={\sum }_{\gamma }{z}_{\gamma }{e}^{-i\gamma u}$ is locally in $L^2$. The theorem retains two alternatives: a nonzero synthesized test with strictly negative complete Weil energy, or $\lambda =0$ with that test identically zero. Nonzero coefficient mass does not exclude a zero synthesis on the chosen observation window.

The corollary on p.224 consequently says that positivity for every smooth compact test in a fixed multiplicative interval union implies one of: RH; infinitely many off-critical zeros; or an ${\ell }^2$ linear dependence of the $x^{-\rho }$ there with at least half the coefficient mass on the off-critical set. It does not eliminate the third alternative or prove that finitely many off-critical zeros would imply none.

Application to the project’s index estimate

The project’s full mixed Gram and common-support negative family already handle actual compact even tests for a specified finite separated off-line orbit frame, including the complete summable zero tail. These interfaces should be reused. Their frame-dependent finite realization supplies no lower growth rate for the number of negative directions as the support grows.

There is also a direct finite-rank upper implication of the complete spectral pairing, separate from passing a finite matrix to its limit. If the actual zero set had only $k<\infty$ distinct nonreal conjugate pairs, let $f$ be the compact-test root variable in $T[f\ast f^{\ast }]$ and put $u_j=\hat{f}({\gamma }_j)$, $v_j=\hat{f}({\overline{\gamma }}_j)$. The $j$th paired term is $2m_j\mathrm{Re}(u_j\overline{v_j})$, with equal positive multiplicities $m_j>0$ at the two conjugate nodes. Here $j$ runs over all unordered conjugate pairs in the full spectral set: the reflected pair $\{-{\gamma }_j,-{\overline{\gamma }}_j\}$ is counted separately when it is distinct. No evenness or reflection quotient is imposed in this argument. For an even root on a four-point orbit, the two pair contributions combine to $4m_j\mathrm{Re}(u_j\overline{v_j})$, consistent with the existing four-point orbit identity. Imposing the $k$ complex-linear conditions $u_j=v_j$ makes these terms $2m_j|u_j{|}^2\ge 0$ and leaves the whole critical-line sum with its positive weights. The antisymmetric pair-evaluation map is consequently injective on every strictly negative complex subspace of compact-test roots, so its complex dimension is at most $k$. This does not posit an actual exceptional zero configuration. Transport to a completed physical operator requires form-core identification and continuity of these evaluations in its form-domain topology.

Under that domain identification, this conditional bounded index is consistent with the complete-form Fourier estimate, whose upper exponential rate is $1/2$, even when $k>0$. The estimate alone therefore does not exclude this conditional case. This is a scope check using existing spectral structure, not a claim that such zeros exist, that they are possible for the actual zeta function, or that no additional theorem could exclude them.

The remaining RH obligation is a sign certificate excluding the residual negative directions, with the original source and domain preserved. Neither the finite inertia count nor this source comparison supplies that certificate or the same-integer Robin signed estimate.