Spectral Oscillations and Jump Dynamics: A Refined Path Toward the Riemann Hypothesis

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Introduction

The distribution of prime numbers remains one of the most profound mysteries in analytic number theory. Central to this mystery is the Riemann Hypothesis (RH), which asserts that all non-trivial zeros of the Riemann zeta function, ζ(s), lie on the critical line Re(s) = 1/2. The implications of RH are far-reaching, primarily providing the tightest possible bound for the error term in the Prime Number Theorem. If RH is true, the difference between the prime-counting function π(x) and the logarithmic integral li(x) is constrained by a factor proportional to x^1/2 log x.

However, the relationship between π(x) and li(x) is not simple. While Gauss famously conjectured that li(x) > π(x) for all x, Littlewood proved in 1914 that the difference li(x) - π(x) changes sign infinitely often. The search for the first such sign change, known as the Skewes number, has led to significant computational and theoretical efforts. The paper arXiv:hal-00627233 enters this discourse by proposing a refined analysis of the jumps of the function li[ψ(x)] - π(x), where ψ(x) is the second Chebyshev function.

The core contribution of arXiv:hal-00627233 is the introduction of a weighted sum, η_N(x), which utilizes the Moebius function μ(n) to construct a more precise approximation of π(x). By studying the jump discontinuities of this function at prime powers, the author provides a new lens through which to view the oscillations of the prime distribution. This article explores the mathematical foundations of this approach, analyzes the behavior of these jumps, and proposes new research pathways that connect these local discontinuities to the global distribution of the zeros of ζ(s).

Mathematical Background

To understand the refinements proposed in arXiv:hal-00627233, we must first define the primary arithmetic functions involved. The prime-counting function π(x) counts the number of primes p ≤ x. The logarithmic integral is defined as li(x) = ∫ dt/log t. The first and second Chebyshev functions are defined as:

• θ(x) = sum of log p for all primes p ≤ x
• ψ(x) = sum of log p for all prime powers p^k ≤ x

The relationship between ψ(x) and the zeros of the zeta function is given by the explicit formula: ψ(x) = x - ∑ (x^ρ/ρ) - log(2π) - (1/2)log(1 - x^-2), where ρ = β + iγ are the non-trivial zeros of ζ(s). Under the Riemann Hypothesis, β = 1/2 for all ρ, leading to the oscillation of ψ(x) around the identity function x with a magnitude of roughly x^1/2.

The source paper focuses on the function ε_ψ(x) = li[ψ(x)] - π(x). This function is particularly interesting because ψ(x) captures information about prime powers, whereas π(x) only counts primes. The author further refines this into the function η_N(x):

η_N(x) = ∑_n=1^N (μ(n)/n) li[ψ(x)^1/n] - π(x)

The introduction of ψ(x) inside the logarithmic integral in η_N(x) creates a hybrid object that tracks the jumps of the Chebyshev function against the discrete steps of the prime-counting function. The author specifically investigates the jumps of η_N(x) at prime powers, asserting that these jumps provide a control mechanism for the sign of the error term.

Main Technical Analysis

Jump Dynamics and Prime Power Discontinuities

The primary technical innovation in arXiv:hal-00627233 is the localized study of jumps at prime powers p^l. A jump of a function f(x) at a point a is defined as the limit of f(a + h) - f(a - h) as h approaches zero. For η_N(x), jumps occur whenever ψ(x) or π(x) changes value.

The author observes that the jumps of li[ψ(p)] - li[ψ(p-1)] are related to the behavior of "Chebyshev primes." Specifically, the statement in arXiv:hal-00627233 regarding the condition η_N(p^l) - η_N(p^l-1) - 1/l > 0 implies a specific growth rate for the logarithmic integral relative to the discrete step of the prime-counting function. As p approaches infinity, the difference η_N(p^l) - η_N(p^l-1) - 1/l is conjectured to approach zero.

This suggests that the energy of the jump at a prime power is almost exactly compensated by the derivative of the logarithmic integral. When this balance is disrupted, it indicates an oscillation in the prime distribution that can be traced back to the zeros of ζ(s). The paper labels primes where the jump of li[ψ(p)] is less than 1 as Chebyshev primes, denoted Ch_n. These primes are critical because they represent locations where the smooth approximation li[ψ(x)] lags behind the actual count of primes.

Spectral Influence on Oscillations

The behavior of η_N(x) is not merely a local phenomenon but is driven by the spectrum of the zeta zeros. By substituting the explicit formula for ψ(x) into the definition of η_N(x), we see that ψ(x)^1/n is approximately equal to (x - ∑ x^ρ/ρ)^1/n. For n=1, the term μ(1)/1 li[ψ(x)] dominates. The terms for n ≥ 2 in the sum for η_N(x) act as filters that remove the secondary biases in the prime distribution (the biases toward p², p³, etc.). By subtracting (μ(n)/n) li[ψ(x)^1/n], the function η_N(x) effectively whitens the error term.

Data presented in arXiv:hal-00627233 illustrates the maximum values of η_N and the corresponding x values. For instance, at N=3, x_max is 6889 with η_max = 1.118. As N increases to 50, the values of η_max and x_max fluctuate, reflecting the inclusion of more Moebius-weighted terms. The persistence of these maxima suggests that even after filtering out the primary biases, significant oscillations remain. Under the Riemann Hypothesis, these oscillations must stay within the bounds of x^1/2+ε. If any η_N(x) were to grow faster than this, it would imply the existence of zeros with Re(ρ) > 1/2.

Novel Research Pathways

1. Moebius-Weighted Kernel Density Estimation

The function η_N(x) can be viewed as a specific type of kernel density estimator for the zeros of the zeta function. A promising research direction would be to generalize the weighting scheme. Instead of using μ(n)/n, one could employ a sequence of weights w_n optimized to minimize the variance of the error term.

• Formulation: Define Ω(x; W) = ∑ w_n li[ψ(x)^1/n] - π(x).
• Methodology: Determine if there exists a weight vector W that isolates the contribution of specific low-lying zeros.
• Connection: If such a spectral filter can be constructed, it would allow for a local test of the Riemann Hypothesis by checking if the filtered signal obeys expected magnitude constraints.

2. The Jump Frequency Conjecture

The source paper discusses the limits of the jumps η_N(p^l) - η_N(p^l-1) - 1/l. A formal investigation into the distribution of these jump magnitudes could yield a new equivalent to the Riemann Hypothesis.

• Formulation: Analyze the sequence of jump residuals δ(p^l) = η_N(p^l) - η_N(p^l-1) - 1/l.
• Methodology: Investigate whether the cumulative sum of these residuals exhibits square-root cancellation.
• Connection: Establish criteria for the location of zeros on the critical line by examining the transformations impact on the residual sum.

3. Chebyshev Prime Density and Critical Line Violations

The third research direction involves establishing a quantitative relationship between the density of Chebyshev primes and the distribution of zeta function zeros. The source paper identifies primes Ch_n where the jump is less than 1 but does not analyze their asymptotic density.

• Formulation: Investigate the density function D(x) = #{p ≤ x : p is a Chebyshev prime}/π(x).
• Methodology: Extend the computational analysis to much larger ranges (up to 10¹²) and analyze the spectral content of deviations.
• Connection: Under RH, D(x) should approach a specific limit. Larger oscillations would indicate zeros off the critical line.

Computational Implementation

(* Section: Refined Chebyshev Jump Analysis *)
(* Purpose: To compute eta_N(x) and visualize its oscillations and jumps *)

Module[{maxVal, nLimit, psi, etaN, primeData, jumps, zeros, gammas},
  maxVal = 2000; (* Range of x to investigate *)
  nLimit = 5;    (* Number of Moebius terms N *)

  (* Define the second Chebyshev function psi(x) using MangoldtLambda *)
  psi[x_] := Total[MangoldtLambda[Range[Floor[x]]]];

  (* Define the refined function eta_N(x) from hal-00627233 *)
  etaN[x_, N_] := Sum[
    (MoebiusMu[n]/n) * LogIntegral[Max[2, psi[x]^(1/n)]], 
    {n, 1, N}
  ] - PrimePi[x];

  (* Generate data for plotting eta_N(x) *)
  primeData = Table[{x, etaN[x, nLimit]}, {x, 2, maxVal, 1}];

  (* Identify the jumps at prime numbers *)
  jumps = Table[
    {p, etaN[p, nLimit] - etaN[p - 0.1, nLimit]}, 
    {p, Prime[Range[PrimePi[maxVal]]]}
  ];

  (* Visualize the refined error function eta_N(x) *)
  Print[ListLinePlot[primeData, 
    Filling -> Axis, 
    PlotLabel -> "Refined Error Function eta_N(x) for N=5",
    FrameLabel -> {"x", "eta_N(x)"}]];

  (* Compare with the first few nontrivial zeta zeros: rho = 1/2 + i*gamma *)
  zeros = Table[ZetaZero[k], {k, 1, 10}];
  gammas = Im[zeros];

  (* Visualize a crude phase driver cos(gamma log x) for x in a range *)
  Print[Plot[Evaluate[Total[Cos[gammas*Log[x]]/gammas]], {x, 10, maxVal},
    PlotLabel -> "Oscillation from first 10 zeta zeros"]];

  (* Analysis of the maximum eta value *)
  Print["Maximum eta_N value in range: ", Max[primeData[[All, 2]]]];
]

Summary of key findings: The investigation of the function η_N(x) and the jumps of li[ψ(x)] provides a sophisticated framework for analyzing prime distribution. By incorporating the Moebius function into the approximation, we move beyond the classical li(x) model to account for the influence of prime powers. The most promising avenue for further research lies in the spectral analysis of the η_N(x) function. If the jumps can be shown to converge at a specific rate, it would provide a new pathway to verify the Riemann Hypothesis over finite intervals. Immediate next steps should involve expanding the calculation of Chebyshev primes to determine if the frequency of negative jumps correlates with the known heights of the zeros on the critical line.

References

• arXiv:hal-00627233: Jumps of the function li[ψ(x)] - π(x) and the Riemann Hypothesis.
• L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II., Math. Comp., 30 (1976).
• H. M. Edwards, Riemann's Zeta Function, Academic Press (1974).