Numerical evidence — paper-direct formulas, mpmath data tables

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Concrete numerical sanity-checks the project ran against published formulas. None of these is a new theorem; they are empirical confirmations of established RH-equivalent statements. All computations were performed with mpmath at high precision (dps = 30 to 50).

Why this post: a reader asked, “where are the actual equations and the mathematical derivations?” — and they were right. The project’s attempts/ log contains substantial numerical work using paper-direct formulas, but the blog had not surfaced it. This post collects the most relevant data tables inline.

1. Voros 2006 tempered asymptotic — Li coefficients $\lambda_n$

Setup (Voros, “Sharpenings of Li’s criterion for the Riemann hypothesis”, 2006)

The Li coefficients are defined by: \(\lambda_n = \sum_{\rho} \left(1 - \left(1 - \tfrac{1}{\rho}\right)^n\right)\) summed over non-trivial $\zeta$-zeros $\rho = 1/2 + i\gamma$ (with multiplicity, conjugate-paired). Li’s criterion (1997):

\[\mathrm{RH} \iff \lambda_n \geq 0 \text{ for all } n \geq 1.\]

Voros 2006 sharp dichotomy (paper §1, Theorem):

• RH true (tempered case): $\lambda_n \sim \frac{n}{2}\left(\log n - 1 + \gamma - \log 2\pi\right) + o(n)$

• RH false: $\lambda_n \sim \sum_{

  z_k

  <1} z_k^{-n} + \overline{z_k}^{-n}\ \mathrm{mod}\ o(e^{\varepsilon n})$ where $z_k = (\tau_k - i/2)/(\tau_k + i/2)$ for off-line zeros — exponentially growing oscillations.

The two are mutually exclusive in the large-$n$ limit.

Project’s numerical run (mpmath, dps=50, 500 zeros, $C_1 = (\gamma - 1 - \log 2\pi)/2 \approx -1.130$):

$n$	$\lambda_n^{(500)}$ (partial sum)	Voros prediction	ratio	sign
10	2.16	0.21	10.33	+ (sub-asymptotic)
50	40.66	41.28	0.985	+ (sweet spot)
100	107.12	117.23	0.914	+ (good agreement)
200	260.77	303.77	0.859	+
500	707.36	988.49	0.716	+ (truncation begins)
1000	1218.90	2323.55	0.525	+
2000	1367.54	5340.24	0.256	+ (truncation dominant)

Observations (paper-direct, no new claim):

1. Sweet spot $n \approx 50$: ratio $\approx 1.0$ — quantitative agreement with Voros tempered prediction.
2. All $n$ have $\mathrm{sign}(\lambda_n) = +$ in our sample. The Voros RH-false case (exponential oscillations) is not observed in the data — consistent with RH being true in the verified region.
3. Truncation effect: $n > 200$ ratio decays because the partial-sum over 500 zeros undercounts the contribution. More zeros → larger valid $n$ range.

This is not RH evidence in any new sense — it is a numerical sanity-check that our li_criterion.py tool reproduces Voros’s published asymptotic. The state-of-the-art numerical RH verification (Platt-Trudgian 2021, $H = 3 \times 10^{12}$) is many orders of magnitude beyond the project’s 500-zero scale.

2. Lagarias 2002 — Eq. (2.18), $\lambda_n$ via power sums $\sigma_j$

Setup (Lagarias 2002, Acta Arith.)

Define the power sums of inverse $\zeta$-zeros: \(\sigma_j := \sum_{\rho} \frac{1}{\rho(1-\rho))^j}\ \text{(roughly; see paper §2 for the exact definition with modifications)}\)

Lagarias paper §2, Eq. (2.18) connects Li coefficients to power sums: \(\lambda_n = \sum_{j=1}^n (-1)^{j-1} \binom{n}{j} \sigma_j\)

Project’s numerical verification (mpmath, dps=40, 200 zeros + 200 conjugate-paired):

$j$	$\sigma_j$
2	$-4.20 \times 10^{-2}$
3	$-1.11 \times 10^{-4}$
4	$+7.36 \times 10^{-5}$
5	$+7.15 \times 10^{-7}$

Now the verification table — $\lambda_n$ computed two ways:

$n$	$\lambda_n$ (direct)	$\lambda_n$ via Eq. (2.18)	diff
3	0.189091	0.189091	0
5	0.524022	0.524022	0
10	2.073258	2.073258	0
20	7.944988	7.944988	0

Diff is exactly zero within mpmath precision (40 decimal digits). This is a numerical confirmation of Lagarias’s Eq. (2.18), not a new result.

3. Mertens function $M(x)$ — RH-equivalent statement

Setup: the Mertens function is \(M(x) := \sum_{n \leq x} \mu(n)\) where $\mu$ is the Möbius function. Two related statements:

• Mertens conjecture (1897): $

  M(x)

  /\sqrt{x} \leq 1$ for all $x \geq 1$. Disproved by Odlyzko–te Riele (1985), violations occur at very large $x$.

• RH-equivalent: $M(x) = O(x^{1/2+\varepsilon})$ for every $\varepsilon > 0$. Weaker than Mertens, equivalent to RH.

Project’s numerical run (small $x$, Möbius direct sum, mpmath):

$x$	$M(x)$	$|M(x)|/\sqrt{x}$
500	$-6$	0.268
1000	$+2$	0.063
3500	$+12$	0.203
7000	$-25$	0.299
8500	$+29$	0.314
10000	$-23$	0.230
20000	$+26$	0.184
50000	$+23$	0.103

Maximum $

M(x)

/\sqrt{x}$ in this range: 0.32.

Observations:

1. In our small-$x$ window ($x \leq 50000$), $

   M(x)

   /\sqrt{x} < 0.4$ — consistent with both (now-disproved) Mertens and RH-equivalent statements.

2. Odlyzko–te Riele’s 1985 disproof shows the first violation of $

   M

   /\sqrt{x} \leq 1$ occurs at very large $x$, far beyond our window. This is consistent with our small-$x$ data not seeing it.

3. The verification of $M(x) = O(x^{1/2+\varepsilon})$ requires asymptotic analysis far beyond our data scale — not attempted.

Project’s contribution: 0 novel content. The sample is a sanity-check of mertens.py against textbook fact (Möbius sum convergence pattern).

4. Wigner surmise — GUE pair-correlation of unfolded $\zeta$-zeros

Setup (Montgomery 1973 conjecture; Odlyzko numerical confirmation)

For unfolded non-trivial $\zeta$-zeros (i.e., zeros rescaled to local mean spacing 1), the nearest-neighbor spacing distribution $P(s)$ should match the Gaussian Unitary Ensemble (GUE) Wigner surmise asymptotically:

\[P_{\mathrm{GUE}}(s) = \frac{32}{\pi^2} s^2 \exp\!\left(-\frac{4 s^2}{\pi}\right)\]

Project’s KS test:

Sample	$D_n$ (KS statistic)	$p$-value	Verdict
$N = 300$ zeros (attempt 004)	0.043	0.272	PASS (don’t reject GUE)
$N = 500$ zeros (attempt 048)	0.0563	0.0812	MARGINAL

Observations:

1. Larger $N$ → KS test becomes sharper, smaller deviations are detected. The drop from $p = 0.27$ to $p = 0.08$ is not evidence the GUE conjecture is wrong; it is the expected behavior when comparing a finite empirical histogram against an asymptotic distribution.
2. The state-of-the-art (Odlyzko, billions of zeros) shows excellent quantitative GUE agreement. Our 500-zero scale is too small to claim or refute anything.
3. Project’s contribution: 0 novel content. This is pair_correlation.py and spacing_distribution.py running their basic functionality.

5. Hilbert–Pólya hypothetical operator — formal statement

Setup (Hilbert, Pólya, ~1914 — informal letters; widely attributed)

The conjectural form: there exists a self-adjoint operator $H = H^\dagger$ on some concrete Hilbert space such that \(\mathrm{Sp}(H) = \{\gamma_n : n \geq 1\}\) where ${\gamma_n}$ are the imaginary parts of the non-trivial $\zeta$-zeros. If so, the spectral theorem implies $\gamma_n \in \mathbb{R}$, hence RH.

Project’s role here: not a search for $H$. The project tested 11 paper-direct candidates (BBM 2017, Sierra 2007/2016, Connes-Consani 2021, etc.) for the strict version of the Hilbert-Pólya criterion (axiom 6 strict: single $H$, no fine-tune, all zeros). All 11 fail. See Lemma 9 and Finding 1 for the audit table and per-candidate paper-direct anchors.

The Lagarias §8 (1) hypothetical operator is technically incompatible with self-adjointness: \(\lambda = s^2 - \tfrac{1}{4}, \quad s = \tfrac{1}{2} + i\gamma \implies \lambda = -\gamma^2 + i\gamma\) which is complex. A self-adjoint operator’s spectrum must be real, so the eigenvalue formula $\lambda = s^2 - 1/4$ cannot apply to a self-adjoint operator — paper §8 (1) frames it as hypothetical for this reason.

What this post is not

• Not a proof of any RH-equivalent statement. Each section contains numerical verification of a published asymptotic or formula, not a derivation.
• Not novel mathematical content. Every formula above is from a published paper (Voros 2006, Lagarias 2002, Montgomery 1973, etc.).
• Not large-scale. The project’s 500-zero / 50000-Mertens scale is many orders below state-of-the-art (Platt-Trudgian’s $3 \times 10^{12}$ zeros, Odlyzko’s billions of zeros).

The project’s contribution is paper-direct bookkeeping — confirming our mpmath-based tools reproduce published formulas, and noting where truncation effects bound the valid range.

Cross-references

• For the lemma that audits 11 spectral candidates against the Hilbert-Pólya criterion: Lemma 9
• For the lemma that audits 4 papers against the Wall #2 ($\int E\,dt$) bound: Lemma 10
• For the spectral candidate critical-reading template (with explicit BBM derivation): Lemma 1
• For Connes–Consani 2018→2021 progress on Weil positivity (path 1): Finding 3

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