Abstract
We compute truncated spectral amplitude sums over the non-trivial zeros of the Riemann zeta function, using 2,001,052 zeros from Odlyzko's tables, and illustrate the recovery of the first zeros by Fourier analysis of \(M(x)/\sqrt{x}\). For the Mertens function the relevant sum is the Odlyzko–te Riele sum \(\sum_{n \le K} 2/|\rho_n\,\zeta'(\rho_n)|\).
The weights are those of the corresponding explicit formulas and \(\zeta'(\rho_n)\) is computed at every zero (true \(|\zeta'(\rho_1)| = 0.79316\)). With these, the Mertens sum crosses \(1\) at \(K = 58\) and \(2\) near \(K = 1{,}040\), and the \(\psi\) sum \(\sum_{n \le K} 2/|\rho_n|\) crosses \(1\) at \(K = 21\); the growth of the Mertens sum is consistent with the conjectural \((\log T)^{5/4}\) law. An amplitude sum exceeding one is not by itself a statement about \(M(x)/\sqrt{x}\): the valid result is Ingham's theorem (1942) with a Fejér kernel, and Odlyzko–te Riele bounded the kernel-weighted sum and aligned the phases by lattice reduction. The divisor error \(\Delta(x)\) has no expansion over zeta zeros, and \(Q(x) - 6x/\pi^2\) lives at scale \(x^{1/4}\).
No classification, prediction, or connection to the Navier–Stokes equations is claimed. Retained: the zero data and the Riemann–von Mangoldt density, the elementary divergence theorem, and the DFT zero-recovery demonstration.
Contents
- Introduction
- Mathematical Framework
- Computation
- Results
- Zero Recovery from Arithmetic Data
- Discussion
- Conclusion
1. Introduction
Background
The explicit formula of analytic number theory expresses arithmetic functions connected to prime distribution as sums over the non-trivial zeros of the Riemann zeta function. For example, the Mertens function \(M(x) = \sum_{n \leq x} \mu(n)\) satisfies
\[M(x) = \sum_{\rho} \frac{x^{\rho}}{\rho\,\zeta'(\rho)} + \text{lower order terms}\]where the sum runs over the non-trivial zeros \(\rho = \tfrac{1}{2} + i\gamma\) of \(\zeta(s)\).
Odlyzko and te Riele (1985) used this formula to disprove the Mertens conjecture (\(|M(x)| < \sqrt{x}\)). Their argument does not rest on the raw amplitude sum \(\sum |1/(\rho\,\zeta'(\rho))|\) alone: it bounds a Fejér-kernel-weighted sum through Ingham's theorem (1942) and aligns the phases \(\gamma_n \ln x\) of the leading terms by lattice reduction, producing explicit \(x\) with \(|M(x)/\sqrt{x}| > 1\).
This Work
We compute the truncated Odlyzko–te Riele sum and the corresponding \(\psi\) sum from a single dataset of 2,001,052 zeta zeros with \(\zeta'(\rho_n)\) evaluated at every zero, record where each crosses \(1\) and \(2\), and illustrate the recovery of the first zeros by Fourier analysis of \(M(x)/\sqrt{x}\).
2. Mathematical Framework
The Explicit Formula
Each arithmetic function \(f\) with an expansion over the zeros of \(\zeta(s)\) has an explicit formula of the form
\[\frac{f(x)}{\sqrt{x}} = \sum_{\rho} w_f(\rho) \cdot x^{i\gamma_\rho} + \text{lower order terms}\]where \(w_f(\rho)\) is the weight of the corresponding explicit formula and \(x^{i\gamma} = e^{i\gamma \ln x}\) is an oscillation at frequency \(\gamma\) in \(\ln x\).
Weights Used
| Function | Weight \(w_f(\rho)\) | Note |
|---|---|---|
| \(M(x) = \sum_{n \leq x} \mu(n)\) | \(\dfrac{1}{\rho\,\zeta'(\rho)}\) | \(\zeta'(\rho_n)\) computed at every zero |
| \(\psi(x) - x\) | \(-\dfrac{1}{\rho}\) | no \(\zeta'\) factor |
| \(\pi(x) - \mathrm{li}(x)\) | \(-\dfrac{\mathrm{li}(x^\rho)}{\ln x}\) | the logarithm is in \(x\), not per zero |
| Divisor error \(\Delta(x)\) | — | no expansion over zeta zeros (Voronoï series is over \(d(n)\)) |
| Squarefree error \(Q(x) - 6x/\pi^2\) | — | lives at scale \(x^{1/4}\), not \(\sqrt{x}\) |
Zero Density
The Riemann–von Mangoldt formula gives the number of zeros with imaginary part up to \(T\):
\[N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T)\]Amplitude Sums
If \(|w_f(\rho_n)| \geq C/\gamma_n^a\) for \(a \leq 1\), then \(\sum_{n \le K} |w_f(\rho_n)| \to \infty\) as \(K \to \infty\), by the Riemann–von Mangoldt density.
Divergence of the amplitude sum does not by itself give a value of \(x\) at which \(|f(x)/\sqrt{x}|\) exceeds a given bound. The valid statement is Ingham's theorem (1942): for a Fejér kernel \(K_T\), the kernel-weighted sum \(\sum_{|\gamma| < T} K_T(\gamma)\, w_f(\rho)\, x^{i\gamma}\) bounds \(\liminf\) and \(\limsup\) of \(f(x)/\sqrt{x}\), and a value of \(x\) is found by aligning the phases \(\gamma_n \ln x\) of the leading terms, which Odlyzko–te Riele did by lattice reduction.
3. Computation
Data: 2,001,052 zeros from Odlyzko's tables, with \(\gamma_1 = 14.1347\) to \(\gamma_{2{,}001{,}052} = 1{,}132{,}490.7\). \(|\zeta'(\rho_n)|\) is computed at every zero (\(|\zeta'(\rho_1)| = 0.79316\)); the measured harmonic means of \(|\zeta'(\rho_n)|\) over blocks of zeros are recorded alongside.
Method: The truncated Odlyzko–te Riele sum \(\sum_{n \le K} 2/|\rho_n\,\zeta'(\rho_n)|\) and the \(\psi\) sum \(\sum_{n \le K} 2/|\rho_n|\) are evaluated at every \(K\) up to 2,001,052, and the first crossings of \(1\) and \(2\) are recorded.
4. Results
Crossings of the Truncated Sums
| Sum | Crosses 1 at | Crosses 2 near | Growth in \(T = \gamma_K\) |
|---|---|---|---|
| Mertens, \(\sum_{n \le K} 2/|\rho_n\,\zeta'(\rho_n)|\) | \(K = 58\) | \(K = 1{,}040\) | consistent with the conjectural \((\log T)^{5/4}\) law |
| \(\psi\), \(\sum_{n \le K} 2/|\rho_n|\) | \(K = 21\) | — | \(\sim \log^2 T\) (elementary, from the zero density) |
Both sums are monotone and unbounded. The Mertens crossing of \(1\) at \(K = 58\) is consistent with Odlyzko–te Riele, whose kernel-weighted argument used the first 2,000 zeros. The crossing is a property of the truncated sum; the passage to a value of \(x\) with \(|M(x)/\sqrt{x}| > 1\) is Ingham's theorem with phase alignment, not the sum alone.
5. Zero Recovery from Arithmetic Data
A discrete Fourier transform on \(M(x)/\sqrt{x}\) sampled at 4,096 log-spaced points up to \(10^9\), without using the explicit formula or computing \(\zeta(s)\), recovers 13 of the first 15 zeta zeros. The phases used in the reconstruction include \(\arg \zeta'(\rho_n)\).
| Rank | Recovered \(\gamma\) | Known \(\gamma\) | Error |
|---|---|---|---|
| 1 | 14.130 | 14.135 | 0.005 |
| 2 | 21.010 | 21.022 | 0.012 |
| 3 | 25.015 | 25.011 | 0.004 |
| 4 | 30.445 | 30.425 | 0.020 |
| 5 | 32.940 | 32.935 | 0.005 |
| 6 | 37.560 | 37.586 | 0.026 |
| 7 | 40.910 | 40.919 | 0.009 |
| 8 | 49.770 | 49.774 | 0.004 |
| 9 | 48.010 | 48.005 | 0.005 |
| 10 | 43.335 | 43.327 | 0.008 |
| 11 | 59.380 | 59.347 | 0.033 |
| 12 | 52.970 | 52.970 | 0.000 |
| 13 | 56.450 | 56.446 | 0.004 |
6. Discussion
Scope
- The amplitude sums are truncated sums over computed zeros; their crossings are recorded as data. Turning a crossing into a statement about \(f(x)/\sqrt{x}\) requires Ingham's kernel-weighted bound and phase alignment, which this computation does not perform.
- Only functions with an expansion over the zeros of \(\zeta(s)\) are treated. The divisor error has no such expansion, and the squarefree error lives at scale \(x^{1/4}\).
- The DFT zero recovery is a demonstration of the explicit formula seen from the arithmetic side; it uses the first zeros only.
- No connection to the Navier–Stokes equations is claimed.
7. Conclusion
- The Mertens sum with computed \(\zeta'(\rho_n)\) crosses \(1\) at \(K = 58\) and \(2\) near \(K = 1{,}040\), with growth consistent with the conjectural \((\log T)^{5/4}\) law; the \(\psi\) sum crosses \(1\) at \(K = 21\).
- The valid bridge from the sum to \(M(x)\) is Ingham's theorem with a Fejér kernel and lattice-reduction phase alignment, as in Odlyzko–te Riele.
- The first zeta zeros are recoverable from \(M(x)/\sqrt{x}\) by a discrete Fourier transform.
The results are computational, grounded in the explicit formula and the Riemann–von Mangoldt density. Every table is reproducible in under 10 seconds on consumer hardware.
Reproducibility
Two independent implementations are provided:
- C:
quick_sum.c(amplitude sums, <1s),paper_data.c(all tables, <2s),explicit_formula.c(point evaluation at any scale),wave_plotter.c(CSV figure data). - Python:
validate.py(reproduces all paper tables),quick_sum.py(core amplitude computation),wave_plotter.py(generates all figures via matplotlib).
Both produce identical results. Zero data: Odlyzko zeros6 (2,001,052 zeros, 36MB).
Code: github.com/senuamedia/lab/tree/main/domains/swt
References
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