Spectral Signals of the Primes: 1/f Noise and Polyphase Sequences in the Riemann Hypothesis

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Introduction

The Riemann Hypothesis remains the most significant unsolved problem in pure mathematics, asserting that all non-trivial zeros of the Riemann zeta function ζ(s) possess a real part equal to 1/2. While traditionally approached through the lens of analytic number theory, recent developments have seen a surge in interdisciplinary methodologies. One of the most intriguing is the connection between the distribution of prime numbers and the theory of signal processing, specifically the study of 1/f noise (flicker noise) and polyphase sequences.

The source paper hal-04209824 provides a novel framework for interpreting the Riemann Hypothesis through the prism of information theory and spectral analysis. The motivation behind this research lies in the Nyman-Beurling criterion, which reformulates the hypothesis as a density problem in a Hilbert space. This suggests that the hypothesis is true if and only if a specific space of functions, related to the fractional part of 1/x, is dense in the space of square-integrable functions.

This article provides a technical analysis of the findings in hal-04209824. We explore the mathematical foundations of the Nyman-Beurling-Baez-Duarte (NBBD) criterion, analyze the spectral properties of the sequences involved, and discuss how the transition from number theory to signal processing offers new tools for attacking the critical line.

Mathematical Background

To understand the arguments presented in hal-04209824, one must establish the properties of the Riemann zeta function ζ(s) and the Mobius function μ(n). The connection between these two is given by the identity 1/ζ(s) = ∑ μ(n)n^-s, valid for the real part of s > 1. The Riemann Hypothesis is equivalent to the statement that the partial sums of the Mobius function, known as the Mertens function, grow no faster than x^(1/2 + ε).

The Nyman-Beurling-Baez-Duarte Criterion

The NBBD criterion provides a Hilbert space formulation of the hypothesis. Let ρ(x) = {x} be the fractional part of x. The hypothesis is true if and only if the distance d_N squared in L^2(0, 1) between the constant function 1 and the subspace spanned by the functions ρ(k/x) approaches zero as N approaches infinity. This distance can be expressed in terms of the Mobius function and the zeta function, creating a direct link between the distribution of primes and the geometry of Hilbert spaces.

Polyphase Sequences and 1/f Noise

A polyphase sequence is a sequence of complex numbers with constant magnitude, typically used in radar and telecommunications for optimal correlation. In signal processing, 1/f noise refers to a signal whose power spectral density is inversely proportional to the frequency. The work in hal-04209824 suggests that the sequence of phases derived from the zeta function on the critical line exhibits 1/f noise characteristics, a property often associated with systems at the edge of chaos.

Main Technical Analysis

Spectral Properties and Zero Distribution

The paper hal-04209824 investigates the distance d_N squared from the Baez-Duarte criterion. The authors highlight that this distance can be approximated by an integral involving the zeta function and a Dirichlet polynomial involving the Mobius function. For the hypothesis to hold, this integral must vanish as N increases. The spectral property here is defined by the behavior of |ζ(1/2 + it)|^-2. Planat and Aschheim observe that the noise generated by the Mobius function acts as a whitening filter for the zeta function's spectral density.

1/f Noise and the Critical Line

The most provocative claim in hal-04209824 is the identification of 1/f noise in the fluctuations of the number of primes. By performing a Fourier transform on the sequence of Mobius values and the logarithmic derivative of the zeta function, the authors demonstrate that the power spectrum follows a power law S(f) proportional to 1/f^α where α is close to 1. This spectral density implies that the zeros are not merely random but are correlated in a way that preserves structural integrity across all scales.

Novel Research Pathways

Quantum Signal Processing (QSP): Use QSP algorithms to simulate the operator (1/ζ) in a quantum system. If the distance d_N can be mapped to a quantum observable, quantum complexity theory could bound the distance from the critical line.
Thermodynamic Formalism: Treat the zeros of the zeta function as a one-dimensional log-gas. A phase transition in this thermodynamic system would correspond to a violation of the hypothesis, allowing researchers to use the Ruelle transfer operator to prove stability.
Polyphase Sequence Synthesis: Construct synthetic L-functions using optimized polyphase sequences and use sieve methods to measure the distance between the Mobius function and these sequences.

Computational Implementation

To demonstrate the spectral properties discussed in hal-04209824, we provide a Wolfram Language implementation that calculates the Baez-Duarte distance and visualizes the phase fluctuations of the zeta function.

(* Section: Spectral Analysis of Zeta and Baez-Duarte Distance *)
(* Purpose: Compute the Baez-Duarte distance and plot phase fluctuations *)

Module[{baezDuarteDist, nMax = 50, tMax = 100},
  (* 1. Define Baez-Duarte coefficients using Mobius sums *)
  baezDuarteDist[n_] := Module[{c, distSq},
    c = Table[Sum[MoebiusMu[j]/j, {j, 1, k}], {k, 1, n}];
    distSq = n^-1 * Sum[(Sum[c[[k]] * FractionalPart[k/m], {k, 1, n}])^2, {m, 1, 1000}];
    distSq
  ];

  (* 2. Output calculation *)
  Print["Baez-Duarte Distance for N=50: ", baezDuarteDist[nMax]];

  (* 3. Plot Phase on the Critical Line *)
  Print[Plot[Arg[Zeta[1/2 + I*t]], {t, 10, tMax}, 
    PlotStyle -> Blue, 
    AxesLabel -> {"t", "Arg(Zeta)"},
    PlotLabel -> "Phase Fluctuations (Polyphase Signal)"]];

  (* 4. Spectral Density Check *)
  Module[{data, psd},
    data = Table[Abs[Zeta[1/2 + I*t]], {t, 10, 500, 0.1}];
    psd = Abs[Fourier[data]]^2;
    Print[ListLogLogPlot[psd[[1 ;; Length[psd]/2]], 
      Joined -> True, 
      PlotLabel -> "Power Spectral Density (1/f Noise Check)"]];
  ]
]

Conclusions

The analysis of hal-04209824 reveals a profound connection between the Riemann Hypothesis and signal processing. By reframing the Nyman-Beurling criterion through the lens of polyphase sequences and 1/f noise, the research provides a new vocabulary for zero distribution. The realization that the Mobius function acts as a spectral whitener suggests that the hypothesis is a fundamental property of information stability in arithmetic systems. Future work should focus on the rigorous computation of the spectral exponent for various L-functions to determine if 1/f noise is a universal feature of the critical line.

References

Planat, M., & Aschheim, R. (2023). The Riemann hypothesis from the viewpoint of the 1/f noise and the polyphase sequences. hal-04209824
Baez-Duarte, L. (2003). A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis.
Beurling, A. (1955). A closure problem related to the Riemann Zeta-function.