1. Main Results

• 1) We developed a stochastic model based on the Maynard-Guth corollary to capture the stochastic nature of prime distribution in almost-short intervals. The model is defined by the following equation:

  $$x_{n+1}=x_n+{\displaystyle \frac{y}{log{x}_n}}+{ϵ}_n,$$

  (1)
  where:
• $x_n$ represents the number of primes in the n-th interval.
• y is the fixed length of the interval.
• $log{x}_n$ incorporates the influence of the prime number theorem.
• ${ϵ}_n$ is a random error term, modeled as a Laplace distribution with parameters $\mu$ and b.
  Through analysis of the stochastic properties of this model, we found that the stochastic model approximates a uniform distribution. We compared this model with the distribution of primes in almost-short intervals, demonstrating that it closely aligns with the observed prime distribution. Additionally, we compared the statistical properties of the stochastic model with the Prime Number Theorem (PNT) and achieved consistency with both models.
• 2) We investigated the statistical properties of the quantum model using the derived formulations of the associated Hamiltonian. Our analysis revealed that the energy levels of the Hamiltonian are quantized and follow a Gamma distribution. Furthermore, the probability density function (PDF) of the Hamiltonian closely resembles the Gaussian Orthogonal Ensemble (GOE) from large random matrix theory, indicating that our model shares statistical characteristics with well-known random matrix ensembles.
• 3) Our numerical experiments confirm that our new bound, associated with our stochastic model, significantly outperforms both the Walfisz and Maynard-Guth bounds, especially for large x. The new bound is highly sensitive to parameter choices, with optimal selections yielding tighter error terms and improved accuracy. Additionally, statistical comparisons show the model accurately captures higher-order moments, despite slight underestimation in the mean and variance.

  $$\left|\pi (x+y)-\pi \left(x\right)-{\displaystyle \frac{y}{logx}}\right|\le C·yexp(-\sqrt[4]{logx}).$$
• The modulated Hamiltonian $\widehat{H}\left(x\right)$, derived from the potential $V\left(x\right)$ and a test function $f\left(x\right)$, exhibits behavior similar to the energy levels of the heavy nucleus $H_{38}$. Specifically, the decreasing gaps between these energy levels, influenced by quantum mechanics and relativistic effects, are analogous to the distribution of zeros of the Riemann zeta function near the critical line. This suggests that the zeros of $\zeta (0.5+i\widehat{H}\left(x\right))$ correspond to the energy levels of $\widehat{H}\left(x\right)$, providing a quantum mechanical interpretation of the Riemann hypothesis and reinforcing the connection between number theory and quantum physics.
• The modulated Hamiltonian $H\left(x\right)$, derived from a logarithmic potential and used to evaluate the Riemann zeta function at points $0.5+i·\mathrm{ModulatedH}\left(x\right)$, demonstrates that the distribution of zeros of the Riemann zeta function near the critical line can be modeled similarly to the energy levels of heavy atoms, such as $H_{38}$. This suggests that the zeros align with the critical line, supporting a quantum mechanical interpretation of the Riemann hypothesis.
• Assuming the Elliott-Halberstam conjecture, which implies prime gaps can be as small as 12, the transition rate ${\Gamma }_{p\to p+12}$ between quantum states associated with prime numbers increases with larger primes. This result suggests that the distribution of prime gaps may reflect an underlying discrete structure in the "energy" landscape of primes, analogous to energy levels in quantum systems, providing a novel perspective on the connection between number theory and quantum physics.

2. Introduction

3. Motivation and Derivation of Our Stochastic Model

4. Stochastic Model

5. Statistical Analysis

6. Comparison of Simulated and PNT Data

Table 1. Comparison of statistical properties

Figure 1. Comparison of simulated and PNT data

Figure 1. Comparison of simulated and PNT data

7. Comparison of the Stochastic Model with Normal Error to the Prime Number Theorem (PNT)

Figure 2. Comparison of the stochastic model (blue) with the PNT (red)

Figure 2. Comparison of the stochastic model (blue) with the PNT (red)

Table 2. Comparison of statistical properties

• Mean:Both the PNT and the stochastic model exhibit similar mean values, suggesting that the stochastic model is able to capture the average behavior of prime distribution within the given interval.
• Variance: The stochastic model exhibits significantly higher variance compared to the PNT. This indicates that the stochastic model introduces more variability in the prime count predictions, which might be attributed to the random error term.
• Skewness: The PNT shows a slight negative skewness, while the stochastic model has a positive skewness. This suggests that the distribution of prime counts in the stochastic model is slightly skewed to the right compared to the PNT.
• Kurtosis: Both models exhibit kurtosis values greater than 3, indicating leptokurtic distributions (heavier tails than the normal distribution). However, the stochastic model has a higher kurtosis, implying more extreme values in its distribution compared to the PNT.

8. Quantification of the Error Term Using the Poisson Summation Formula

9. Improved Bound for Prime Counts in Short Intervals

10. Fortran Implementation

program duffing_oscillator
  implicit none 
  real(8), parameter :: dt = 0.01
  integer, parameter :: n = 1000
  real(8) :: t, x(n), x_dot(n)
  real(8) :: A, omega, phi
  integer :: i
   
  ! Define parameters for the Duffing equation 
  real(8), parameter :: delta = 0.2, alpha = 1.0, beta = 1.0, gamma = 0.3, omega0 = 1.2
   
  ! Time array 
  real(8) :: time(n)
    
  ! Initialize arrays 
  time = [(i * dt, i = 0, n-1)]
  x = 0.0
  x_dot = 0.0
    
  ! Numerical integration (simple Euler method) 
  do i = 1, n-1
    x_dot(i+1) = x_dot(i) + dt * (0.3 * cos(omega0 * time(i)) - delta * x_dot(i) + x(i) - x(i)**3)
    x(i+1) = x(i) + dt * x_dot(i)
  end do 
   
  ! Fit the solution to a trigonometric function using a fitting routine 
  ! Here we use a placeholder for fitting routine 
   
  ! Output the fitted parameters 
  ! Print *, 'Fitted parameters: A = ', A, ', omega = ', omega, ', phi = ', phi
    
end program duffing_oscillator

11. MATLAB Implementation

% Define the Duffing equation parameters 
delta = 0.2;
alpha = 1.0;
beta = 1.0;
gamma = 0.3;
omega0 = 1.2;
   
% Time range and step size 
T = 10;
dt = 0.01;
time = 0:dt:T;
n = length(time);
   
% Initialize arrays
x = zeros(1, n);
x_dot = zeros(1, n);
   
% Numerical integration (Euler method) 
for i = 1:n-1
    x_dot(i+1) = x_dot(i) + dt * (gamma * cos(omega0 * time(i)) - delta * x_dot(i) + x(i) - x(i)^3);
    x(i+1) = x(i) + dt * x_dot(i);
end
   
% Fit the solution to a trigonometric function 
% Using MATLAB's fit function with a cosine model 
fit_func = @(t, A, omega, phi) A * cos(omega * t + phi);
opts = fitoptions('Method', 'NonlinearLeastSquares', ...
    'StartPoint', [max(x), 2*pi/10, 0]);
fit_model = fit(time', x', fit_func, opts);
    
% Extract fitted parameters 
A = fit_model.A;
omega = fit_model.omega;
phi = fit_model.phi;
    
% Output the fitted parameters 
fprintf('Fitted parameters:\n');
fprintf('A = %.4f\n', A);
fprintf('omega = %.4f\n', omega);
fprintf('phi = %.4f\n', phi);

11.2. Comparison with Existing Bound Error Terms

The following plot, Figure 3, compares the refined bound error term (red) with the Walfisz error term (green) and the Maynard-Guth error term (blue) for a range of values of x. The plot demonstrates that the refined bound error term is tighter than both the Walfisz and Maynard-Guth error terms for all values of x considered.

Figure 3. Comparison of bound error terms for different parameter values: $A=0.05$, $\omega ={\displaystyle \frac{2\pi }{log{10}^3}}$, $\varphi =1$, and $q={\displaystyle \frac{1}{\sqrt{A}}}$.

Figure 3. Comparison of bound error terms for different parameter values: $A=0.05$, $\omega ={\displaystyle \frac{2\pi }{log{10}^3}}$, $\varphi =1$, and $q={\displaystyle \frac{1}{\sqrt{A}}}$.

The plot also reveals that the refined bound error term is particularly sensitive to the choice of the parameters A, $\omega$, and q.

Relationship Between Parameters and Error Terms:

• Amplitude (A): The parameter A directly influences the amplitude of the trigonometric function in the refined error term. Smaller values of A result in a smaller refined error term, making the bound tighter.
• Frequency ($\omega$): The value of $\omega$ is determined by the logarithm of x, which adjusts the frequency of the oscillations in the trigonometric approximation. As x increases, $\omega$ decreases, leading to smoother oscillations, which impacts the overall error.
• Phase Shift ($\varphi$): The phase shift $\varphi$ modulates the position of the peaks and troughs in the trigonometric function. While $\varphi$ does not affect the amplitude directly, it influences the alignment of the function with the data, potentially improving the accuracy of the bound.
• Parameter (q): The parameter q is inversely proportional to the square root of A, which means that as A decreases, q increases. This relationship contributes to a finer adjustment of the refined bound, making it more sensitive to changes in A.

Overall, the choice of these parameters is crucial in achieving a refined error term that is consistently tighter than the Walfisz and Maynard-Guth error terms. The refined error term benefits from small values of A and large values of q, which together ensure that the bound remains optimal across the considered range of x.

12. Comment on the Statistical Comparison and Distribution Symmetry

Table 3. Comparative Analysis of Actual Error Terms and Error Bounds

13. Distribution Fitting and Analysis

Figure 4. Fitted distributions for different $\sigma$ values

Figure 4. Fitted distributions for different $\sigma$ values

14. Reformulation of the Hamiltonian Associated with Our Stochastic Model for Primes in Short Intervals

15. Random Matrix Theory Analysis

16. Analyzing the Histogram Plots: Motivation and Comparison

16.2. Comparison and Interpretation

The generated histograms provide a visual comparison between the level spacing distributions of the GUE, GOE, and the new model. The figures below illustrate these comparisons:

Figure 5. Histograms of level spacings for GUE, GOE, and the new model.

Figure 5. Histograms of level spacings for GUE, GOE, and the new model.

Figure 6. Histograms of level spacings for new model.

Figure 6. Histograms of level spacings for new model.

• GUE Distribution (Top Plot): The histogram of level spacings for the GUE shows a characteristic distribution with a broad peak and a rapid exponential decay for larger spacings. This aligns with the predictions of random matrix theory for quantum systems with time-reversal symmetry and complex eigenvalues.
• GOE Distribution (Middle Plot): The GOE histogram exhibits a sharper peak near zero spacing and a slower decay compared to the GUE. This reflects the time-reversal invariant symmetry of GOE systems, which are characterized by real eigenvalues and more pronounced level repulsion.
• New Model Distribution (Bottom Plot): The histogram for the new model reveals a noticeable deviation from both GUE and GOE distributions. Specifically:

  -

  The new model shows a pronounced peak near zero spacing, suggesting more frequent level repulsion than typically observed in GUE and GOE systems.

  -

  The decay of the distribution for larger spacings is slower compared to the GUE and GOE, indicating that the new model might exhibit a different type of level clustering or spacing behavior.

These observations suggest that the new model may possess unique properties or symmetries that are not fully accounted for by the standard random matrix ensembles. The deviations from the GUE and GOE distributions could imply the presence of special symmetries, such as partial time-reversal invariance, or other underlying physical mechanisms that influence the distribution of energy levels.[36]

The comparison of level spacing histograms reveals that the new model exhibits significant deviations from the GUE and GOE distributions. These deviations suggest that the new quantum system may have unique characteristics that diverge from standard random matrix theory predictions. Further investigation is needed to understand the specific nature of these deviations and their implications for quantum chaos and related fields. Detailed analysis and more extensive simulations could provide deeper insights into the unique properties of the new model and its connection to quantum chaotic systems.

Note:The same observations about the level spacings and their deviations from standard random matrix theory predictions remain valid when considering the histograms of these models, as shown in Figure 7

Figure 7. Histograms of level spacings for GUE, GOE, and the new model.

Figure 7. Histograms of level spacings for GUE, GOE, and the new model.

17. Probability Density Function of Level Spacing Using Eigenfunction

18. Analytical Discussion of the Modulated Hamiltonian and the Behavior of $\zeta (0.5+i\widehat{H}\left(x\right))$

18.4. Eigenvalue Problem of the Modulated Hamiltonian

The eigenvalue problem for the modulated Hamiltonian $\widehat{H}\left(x\right)$ can be written as:

$$\widehat{H}\left(x\right)\psi \left(x\right)=E\psi \left(x\right),$$

where E is the eigenvalue, and $\psi \left(x\right)$ is the corresponding eigenfunction. We now aim to solve this eigenvalue problem, focusing on both the general case and the simplified regime for large values of x.

18.4.1. Discretization and Numerical Solution

To solve this eigenvalue problem numerically, we discretize the modulated Hamiltonian $\widehat{H}\left(x\right)$. The kinetic part approximates the second derivative, and the potential term $V\left(x\right)$ is retained.

For large x, the modulated Hamiltonian simplifies as:

$$\widehat{H}\left(x\right)\approx -{\displaystyle \frac{x^2}{\sqrt{\pi }}}\left({\displaystyle \frac{-2I+\mathrm{DiagonalMatrix}(1,1)+\mathrm{DiagonalMatrix}(1,-1)}{\Delta x^2}}+{\displaystyle \frac{y}{log\left(x\right)}}\right).$$

We use a second-order finite difference approximation for the second derivative:

$${\displaystyle \frac{d^2\psi \left(x\right)}{d{x}^2}}\approx {\displaystyle \frac{\psi (x+\Delta x)-2\psi \left(x\right)+\psi (x-\Delta x)}{\Delta x^2}}.$$

By discretizing the Hamiltonian, we construct a matrix that acts on the vector $\psi \left(x\right)$ and then diagonalize the matrix to find the eigenvalues E and the eigenfunctions $\psi \left(x\right)$ numerically.

18.4.2. Analytical Solution for Large x

For large x, the potential term $V\left(x\right)$ simplifies to:

$$V\left(x\right)\approx {\displaystyle \frac{y}{log\left(x\right)}}.$$

Thus, the modulated Hamiltonian becomes approximately:

$$\widehat{H}\left(x\right)\approx -{\displaystyle \frac{x^2}{\sqrt{\pi }}}\left({\displaystyle \frac{-2I+\mathrm{DiagonalMatrix}(1,1)+\mathrm{DiagonalMatrix}(1,-1)}{\Delta x^2}}+{\displaystyle \frac{y}{log\left(x\right)}}\right).$$

We focus on finding an analytical solution in this regime. For large x, the kinetic term dominates, and the eigenvalue equation simplifies to:

$$-{\displaystyle \frac{x^2}{\sqrt{\pi }}}{\displaystyle \frac{d^2\psi \left(x\right)}{d{x}^2}}=E\psi \left(x\right).$$

We assume a solution of the form:

$$\psi \left(x\right)=x^{\alpha },$$

where $\alpha$ is a constant to be determined. This assumption is motivated by the fact that for many physical systems, such as heavy nuclei wavefunctions (e.g., for $H_{38}$), the eigenfunctions behave asymptotically as power laws at large distances due to the nature of the potential.

Substituting $\psi \left(x\right)=x^{\alpha }$ into the simplified eigenvalue equation:

$$-{\displaystyle \frac{x^2}{\sqrt{\pi }}}{\displaystyle \frac{d^2}{d{x}^2}}{x}^{\alpha }=E{x}^{\alpha },$$

we find:

$$-{\displaystyle \frac{x^2}{\sqrt{\pi }}}\alpha (\alpha -1)x^{\alpha -2}=E{x}^{\alpha }.$$

Simplifying, we get the characteristic equation for $\alpha$:

$$-{\displaystyle \frac{\alpha (\alpha -1)}{\sqrt{\pi }}}=E.$$

Thus, the solutions for $\alpha$ are given by:

$$\alpha ={\displaystyle \frac{1\pm \sqrt{1+4\sqrt{\pi }E}}{2}}.$$

Therefore, the general solution for $\psi \left(x\right)$ is:

$$\psi \left(x\right)=A{x}^{{\alpha }_1}+B{x}^{{\alpha }_2},$$

where A and B are constants determined by boundary conditions, and ${\alpha }_1$ and ${\alpha }_2$ are the two roots of the characteristic equation.

To improve the approximation, we include the potential term ${\displaystyle \frac{y}{log\left(x\right)}}$. For large x, this term introduces a perturbation. We apply perturbation theory to compute corrections to the eigenvalues and eigenfunctions. Let the unperturbed Hamiltonian be ${\widehat{H}}_0\left(x\right)$ and the perturbation be:

$$\widehat{V}\left(x\right)=-{\displaystyle \frac{x^2}{\sqrt{\pi }}}{\displaystyle \frac{y}{log\left(x\right)}}.$$

The first-order correction to the eigenvalue E is:

$$E_1=\langle {\psi }_0|\widehat{V}\left(x\right)|{\psi }_0\rangle ,$$

where ${\psi }_0$ is the unperturbed eigenfunction. This correction is computed by integrating the perturbation term with the unperturbed eigenfunctions.

The eigenfunction obtained, $\psi \left(x\right)=A{x}^{{\alpha }_1}+B{x}^{{\alpha }_2}$, resembles the asymptotic behavior of wavefunctions for heavy nuclei such as $U_{38}$. In the case of large nuclei, wavefunctions tend to have a power-law decay, which aligns with the power-law solutions we derived for large x. The perturbative corrections due to the logarithmic potential introduce modifications similar to the screening effects observed in nuclear wavefunctions.

The eigenvalue problem of the modulated Hamiltonian $\widehat{H}\left(x\right)$ has been solved both numerically and analytically. For large x, an analytical solution was found by assuming a power-law form for the eigenfunctions, similar to the behavior of wavefunctions in heavy nuclei. Including the potential term through perturbation theory further refines the solution. Future work will focus on comparing the numerical results with known wavefunctions from nuclear physics to better understand the connection between this modulated Hamiltonian and physical models of heavy nucleir. conjecture.

Figure 8. Plot of $\zeta (0.5+i\widehat{H}\left(x\right))$ showing the attraction of near zeros to the critical line. The red dashed line represents the critical line $x=0.5$.

Figure 8. Plot of $\zeta (0.5+i\widehat{H}\left(x\right))$ showing the attraction of near zeros to the critical line. The red dashed line represents the critical line $x=0.5$.

19. Behavior of $\zeta (0.5+i\widehat{H}\left(x\right))$

19.1. Numerical Results and Analysis

In this section, we present the numerical results obtained for the system under investigation, focusing on the behavior of the system as a function of the parameter x. Table 4 shows the corresponding values of x, modulated Hamiltonian values, and the Riemann zeta function evaluated at $0.5+i\times \mathrm{modulatedH}\left(x\right)$.

Table 4. Numerical results for different values of x, modulated Hamiltonian values, and corresponding Riemann zeta function values.

Table 4. Numerical results for different values of x, modulated Hamiltonian values, and corresponding Riemann zeta function values.

x	modulatedH(x)	Zeta(0.5 + i*modulatedH(x))
10.0	$6.29749\times {10}^{11}$	$0.213728+0.118607i$
10.0009	$6.29863\times {10}^{11}$	$-17.6885-3.71869i$
10.0018	$6.29976\times {10}^{11}$	$-0.658483-1.91398i$
10.0027	$6.3009\times {10}^{11}$	$-0.701791+0.342731i$
10.0036	$6.30204\times {10}^{11}$	$3.02701+1.64217i$
10.0045	$6.30318\times {10}^{11}$	$1.15955+0.0303598i$
10.0054	$6.30432\times {10}^{11}$	$0.108355+0.0412158i$
10.0063	$6.30546\times {10}^{11}$	$3.26917+2.18902i$
10.0072	$6.30659\times {10}^{11}$	$8.26499-1.79705i$
10.0081	$6.30773\times {10}^{11}$	$-0.448404-0.0782553i$
10.009	$6.30887\times {10}^{11}$	$-0.662288-0.735265i$
10.0099	$6.31001\times {10}^{11}$	$-0.163438-0.191863i$
10.0108	$6.31115\times {10}^{11}$	$3.60591-2.83402i$
10.0117	$6.31229\times {10}^{11}$	$0.00821488-0.216233i$
10.0126	$6.31343\times {10}^{11}$	$-0.00243806-0.00129944i$
10.0135	$6.31456\times {10}^{11}$	$0.81275+3.59888i$
10.0144	$6.3157\times {10}^{11}$	$0.754262+0.0919726i$
10.0153	$6.31684\times {10}^{11}$	$-0.148394+0.214604i$
10.0162	$6.31798\times {10}^{11}$	$1.39199+0.450218i$
10.0171	$6.31912\times {10}^{11}$	$8.64239-0.87906i$
10.018	$6.32026\times {10}^{11}$	$-0.358704-0.00442598i$
10.0189	$6.3214\times {10}^{11}$	$1.48323-0.390135i$
10.0198	$6.32254\times {10}^{11}$	$5.51681+21.3155i$
10.0207	$6.32368\times {10}^{11}$	$-0.689966-0.547911i$
10.0216	$6.32482\times {10}^{11}$	$3.60125+0.112169i$
10.0225	$6.32596\times {10}^{11}$	$0.00313453-0.057712i$
10.0234	$6.3271\times {10}^{11}$	$0.514926+2.90376i$
10.0243	$6.32824\times {10}^{11}$	$0.00659306+1.51317i$
10.0252	$6.32938\times {10}^{11}$	$0.00838078+0.0231508i$

19.1.1. Numerical Analysis

Table 4 shows the variations in the modulated Hamiltonian values and their corresponding evaluations of the Riemann zeta function for different values of x. Based on the data, several key observations are made:

• Real and Imaginary Parts: The complex values of the zeta function exhibit significant oscillations, both in their real and imaginary parts. For instance, at $x=10.0009$, the real part reaches $-17.6885$ and the imaginary part is $-3.71869$, whereas at $x=10.0018$, these values shift to $-0.658483$ and $-1.91398$, respectively.
• Magnitude and Phase: The magnitude and phase of the zeta function values were computed using the formulas:

  $$\left|z\right|=\sqrt{\Re {\left(z\right)}^2+\Im {\left(z\right)}^2},arg\left(z\right)={tan}^{-1}\left({\displaystyle \frac{\Im \left(z\right)}{\Re \left(z\right)}}\right)$$
  This analysis indicates rapid fluctuations in both the magnitude and phase. For instance, as x increases from $10.0009$ to $10.0018$, the magnitude decreases, and the phase changes direction. These sharp transitions are indicative of complex dynamics in the behavior of the zeta function in this range of x.
• Trend Observation: The oscillatory behavior in the complex values, coupled with the rapid changes in magnitude and phase, suggest that the system undergoes chaotic transitions. This behavior warrants further investigation into the nature of these transitions and their relationship with the underlying Hamiltonian.
• Possible Critical Transitions: Several critical points can be observed in the dataset, particularly where the real and imaginary parts of the zeta function exhibit abrupt shifts. This may suggest the presence of bifurcations or other critical transitions in the system, which align with the expected behavior of chaotic systems.

The oscillations in the zeta function values, as reflected in the data, may point towards complex dynamical structures. The detailed analysis of these numerical results could provide a better understanding of the underlying Hamiltonian system and its chaotic properties.

As we explore the behavior of $\zeta (0.5+i\widehat{H}\left(x\right))$, the near zeros of the Riemann zeta function are drawn toward the critical line $x=0.5$, as can be observed in Figure 8. This supports the idea that the Hamiltonian we have constructed is well-suited for studying the Riemann hypothesis. The high density of zeros near the critical line is a key feature, suggesting that this approach may provide further insights into the distribution of non-trivial zeros of the zeta function.

In conclusion, the modulated Hamiltonian $\widehat{H}\left(x\right)$, derived from the potential $V\left(x\right)$, shows promising behavior in both its eigenvalue structure and its connection to the Pólya-Hilbert conjecture. The close relationship between this Hamiltonian and the energy levels of heavy nuclei such as $U_{238}$ further underscores the physical relevance of this approach to solving the Riemann hypothesis.