How does the oscillatory structure interact with the damping factor in a function combining log, sine, and exponential, and what cumulative behavior results from this interaction?
Definition of \(Z(x;21)=x^{-5}\ln x\sin(2\pi/x)e^{-x/21}\); analysis of its sign and roots (\(x=1\), \(x=2\)); study of its accumulation \(A(x)\) (which attains its global minimum at \(x=2\)). Derivation of the asymptotic envelope for the real function \(Z(x) \sim \frac{2\pi\ln x}{x^6}e^{-x/21}\), in addition to the asymptotic envelope of the complex extension \(K(x_p)\ln t/t^6\).
Analytic proof of the sign and roots of \(Z\); analytic proof of \(A_\infty<0\) via two standard inequalities (reviewed and confirmed); separate numerical verification (Appendix D) to check implementation and numerical consistency.
Use the slider below to change the cutoff \(x_p\). The graph's range is extended to \(x=50\) to confirm that \(A(x; x_p)\) continues to increase and does not peak at \(x=42\).
Definition. Let \(\gamma_2 \approx 21.022039\) be the ordinate of the second positive nontrivial zero of the Riemann zeta function. We define the model parameter:
Naming note. The choice \(x_p=21\) is a free modelling choice, inspired by \(\lfloor \gamma_2 \rfloor\) for numerical approximation purposes only. It is not a derivation from the zeta function, and the model makes no claim of a structural mathematical relationship between Riemann zeros and the function \(Z\).
For \(x_p>0\), the real-variable function satisfies:
For \(x>1\), the factors \(x^{-5}\), \(\ln(x)\), and \(e^{-x/x_p}\) are strictly positive. The sign is determined by \(\sin(2\pi/x)\). If \(1 < x < 2\), then \(\pi < 2\pi/x < 2\pi\), so the sine is negative. If \(x > 2\), then \(0 < 2\pi/x < \pi\), so the sine is positive. At \(x=1\), \(\ln x = 0\); at \(x=2\), \(\sin(2\pi/x)=0\). These are the only two roots in \([1,\infty)\).
Define \(A(x):=\int_1^x Z(t;21)\,dt\). By the Fundamental Theorem of Calculus, \(A'(x) = Z(x;21)\). Since \(Z(x)<0\) on \((1,2)\) and \(>0\) on \((2,\infty)\), \(A(x)\) decreases on \((1,2)\), attains a global minimum at \(x=2\), and increases on \((2,\infty)\).
The integral \(A_\infty := \int_1^\infty Z(t;21)\,dt\) converges absolutely because of the factor \(e^{-t/21}\). Substituting \(t = 2\pi/x\):
Splitting the integral over \((0,\pi)\) and \((\pi, 2\pi)\) and substituting \(t \mapsto 2\pi - t\) in the second part:
Convention: In this proof, we use \(v\) (instead of \(A\)) for the auxiliary variable \(v = 2\pi/(\pi+u)\), to avoid confusion with the accumulation function \(A(x)\).
Step 1. Let \(\varphi(u) := \log W(\pi+u) - \log W(\pi-u)\) on \((0,\pi)\). We want to show \(\varphi(u) > 0\), i.e. \(W(2\pi-t) > W(t)\) with \(t=\pi-u\).
Step 2. \(\lim_{u\to 0^+}\varphi(u)=0\). It suffices to show \(\varphi'(u) > 0\). Differentiating \(\log W\):
where \(v = 2\pi/(\pi+u)\), \(w = 2\pi/(\pi-u)\), and \(g(s) = 3 - 1/\ln s\).
Step 3. Multiplying by \(2\pi\) and substituting (so that \(w = v/(v-1)\)):
For \(u\in(0,\pi)\), \(v = 2\pi/(\pi+u)\) lies strictly inside \((1,2)\).
Step 4 (positivity of \(E(v)\) on \((1,2)\)). Two standard inequalities:
Substituting:
On \((1,2)\), \(\dfrac{5v-1}{2} > 2\) (strict because \(v>1\); equality at \(v=1\) lies outside), and \(2 > \dfrac{1}{\ln 2}\approx 1.4427\), so the first bracket is positive. The second term is always positive. Hence \(E(v) > 0\) on all of \((1,2)\).
Step 5. Since \(v(u)\) remains inside \((1,2)\) on all of \((0,\pi)\), positivity of \(E(v)\) on \((1,2)\) gives \(\varphi'(u) > 0\). With \(\lim_{u\to 0^+}\varphi(u)=0\), we obtain \(\varphi(u) > 0\) for every \(u\in(0,\pi)\), i.e. \(W(2\pi-t) > W(t)\) on \((0,\pi)\).
Step 6. In the symmetric form:
Since \(W(2\pi-t) > W(t)\) and \(\sin t > 0\), the integrand is negative pointwise, hence:
Note. The above establishes \(A_\infty<0\) analytically and does not require numerical verification. The numerical verification in Appendix D is presented as a separate check of the implementation, not as a substitute for the proof.
Result. Since \(A(2)<0\), \(A'(x) = Z(x;21) > 0\) on \((2,\infty)\), and \(A_\infty < 0\):
To examine behaviour along the critical line \(s=\tfrac12+it\), define an explicit complex extension using the principal branch of the logarithm \(\operatorname{Log}(s)\):
As \(t \to \infty\), the asymptotic envelope for the complex extension is given by:
Principal branch. \(\operatorname{Log}(s) = \ln|s| + i\arg(s)\) with \(-\pi < \arg(s) \le \pi\).
Asymptotic equivalence. We write \(f(t) \sim g(t)\) iff \(\lim_{t\to\infty} f(t)/g(t) = 1\).
Derived result. The asymptotic envelope follows from the explicit complex extension, and describes the \(t\to\infty\) behaviour. It does not constitute a result about the Riemann zeta function itself.
Source of the target value. The value \(K \approx 6.1353\) is \(K(21) = 2\pi e^{-1/42}\) itself, computed from the model's own defining formula. Therefore the "agreement" between \(K(21)\) and \(6.1353\) is circular by construction, not an independent test.
The table below is best read as a sensitivity analysis of the choice of \(x_p\) over a bounded range (19–22), not as evidence that 21 is uniquely favoured. The comparison is informative about the curvature of \(K(x_p)\) around the chosen value, not about its uniqueness.
| Candidate \(n\) | \(K(n) = 2\pi e^{-1/(2n)}\) | Diff \(|K(n)-6.1353|\) |
|---|---|---|
| 19 | 6.119995 | 0.015305 |
| 20 | 6.128053 | 0.007247 |
| 21 | 6.135353 | 0.000053 |
| 22 | 6.141996 | 0.006696 |
In earlier drafts of the project, a numerically interesting coincidence was noted. If we define the roots \(t_k\) as those satisfying \(Z'(t_k) = 0\) (roots of the function \(Z\), not Riemann zeros), then evaluating the second derivative of the function \(Z\) — i.e. \(Z''(t_k)\), not a "second derivative of a physical potential" — at the fifth root \(t_5 \approx 32.935\) was reported to produce a very small numerical value of order \(10^{-123}\).
Terminological and numerical note. \(Z(x)\) is a purely mathematical model function, not a physical potential. The value \(Z''(t_5)\) has not been re-verified numerically in this version, and should be recomputed using the function \(Z\) defined in §1 before being treated as a numerical result. As for the comparison with the order of \(\Lambda \sim 10^{-122} M_{Pl}^2\), it is an exponential comparison only, not a physical comparison in unified units, until normalizations and units are specified.
A bolder philosophical hypothesis (Jabri-C) proposes that spacetime is not an independent stage but the "output" of a mathematical code. This is formalized hypothetically via an action linking Einstein's geometry to the field \(Z(x)\):
To ensure reproducibility of the results and to verify the derivatives, the following tools are provided:
Asymptotic formula: \(|\mathcal{Z}(1/2+it;21)| \sim K(21)\cdot\ln(t)/t^6\), where \(K(21) = 2\pi e^{-1/42} \approx 6.135353\).
Measured quantity: \(C(t) := |\mathcal{Z}(1/2+it;21)|\cdot t^6/\ln(t)\); expected \(C(t)\to K(21)\) as \(t\to\infty\).
Error definitions: "Absolute error" is \(|C(t)-K(21)|\); "relative error" is \(|C(t)-K(21)|/K(21)\).
| \(t\) | \(K(21)\cdot\ln(t)/t^6\) (asymptotic) |
\(|\mathcal{Z}(1/2+it;21)|\) (numerical — §8) |
\(C(t)\) | \(|C(t)-K(21)|\) (absolute) |
\(|C(t)-K(21)|/K(21)\) (relative) |
|---|---|---|---|---|---|
| \(10^2\) | \(2.8254\times10^{-11}\) | \(2.9860\times10^{-11}\) | \(6.484090\) | \(0.348738\) | \(5.68\times10^{-2}\) |
| \(10^3\) | \(4.2382\times10^{-17}\) | \(4.3463\times10^{-17}\) | \(6.291918\) | \(0.156565\) | \(2.55\times10^{-2}\) |
| \(10^4\) | \(5.6509\times10^{-23}\) | \(5.7325\times10^{-23}\) | \(6.223935\) | \(0.088583\) | \(1.44\times10^{-2}\) |
| \(10^5\) | \(7.0636\times10^{-29}\) | \(7.1290\times10^{-29}\) | \(6.192194\) | \(0.056842\) | \(9.26\times10^{-3}\) |
| \(t\to\infty\) | \(0\) | \(0\) | \(K(21)=6.135353\) | \(0\) | \(0\) |
Note on the final row: The 0 values in the "asymptotic expectation" and \(|Z|\) columns mean that as \(t \to \infty\), they approach zero, not that they have a finite value of zero.
Asymptotic Convergence. As \(t\to\infty\):
Thus, the leading relative deviation from \(K(21)\) decays as \(1/(\ln t)^2\). This asymptotic expansion is derived analytically, explaining the slow convergence seen in the table.
This appendix presents the details of the numerical verification (separate from the analytic proof) that supports the results, without replacing the proofs:
For all \(t \ge 2\), the bound \(|\sin(2\pi/t)| \le 2\pi/t\) holds. Since \(\ln t > 0\) on this domain,
To bound the integral, note that \(g(t)=t^{-6}\ln t\) is decreasing for \(t\ge2\), since
Therefore, for every \(X\ge2\),
At \(X=100\):
import numpy as np
from scipy.integrate import quad
# D.1 — evaluate A(100) and the tail bound
def Z_real(t, xp=21.0):
if t <= 1: return 0.0
return (t**-5) * np.log(t) * np.sin(2*np.pi/t) * np.exp(-t/xp)
A_100, err_A100 = quad(Z_real, 1, 100, limit=500,
epsabs=1e-15, epsrel=1e-12)
X = 100.0
tail = 42*np.pi * X**-6 * np.log(X) * np.exp(-X/21.0)
print(f"A(100) = {A_100:.12f} (quad err: {err_A100:.1e})")
print(f"Tail bound = {tail:.3e}")
print(f"A_inf < 0 (numerically supported): {A_100 + err_A100 + tail < 0}")
print()
# D.2 — evaluate |Z(1/2+it;21)|
def Z_cplx(s, xp=21.0):
return (s**-5) * np.log(s) * np.sin(2*np.pi/s) * np.exp(-s/xp)
K21 = 2*np.pi*np.exp(-1/42)
print(f"K(21) = 2*pi*e^(-1/42) = {K21:.6f}")
print(f"{'t':>8} {'K ln(t)/t^6':>16} {'|Z|':>16} "
f"{'C(t)':>10} {'|C-K|':>10} {'rel':>10}")
print("-" * 80)
for t in [1e2, 1e3, 1e4, 1e5]:
z = abs(Z_cplx(0.5 + 1j*t))
C = z * t**6 / np.log(t)
err = abs(C - K21); rel = err / K21
asymp = K21 * np.log(t) / t**6
print(f"{t:>8.0e} {asymp:>16.4e} {z:>16.4e} "
f"{C:>10.6f} {err:>10.6f} {rel:>10.2e}")Numerical evaluation and tail bound. The numerical integration gave \(A(100) \approx -0.0246463895708\), and scipy.integrate.quad reported an error estimate of approximately \(4.3 \times 10^{-16}\) in the reference execution. The analytic tail bound at \(X=100\) is less than \(5.2 \times 10^{-12}\).
The error estimate returned by quad is not a rigorous guaranteed bound; therefore, these results are presented as a strong numerical verification, not a formal computational certificate. Since the sign \(A_\infty < 0\) has already been proven analytically in §3, the final result does not rely on assuming the integration error estimate is a rigorous bound.
scipy.integrate.quad with epsabs=1e-15, epsrel=1e-12, limit=500.numpy.log uses the principal branch with imaginary part in \((-\pi, \pi]\), the convention of §4.