The Arithmetic of Equidistribution: Linking Farey Sequences to the Riemann Hypothesis

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Introduction

The distribution of rational numbers in the unit interval is a fundamental problem in number theory with deep implications for the Riemann Hypothesis (RH). The source paper arXiv:hal-00353820v1 explores the intricate relationship between Farey sequences, Riemann sums, and the Mertens function. Historically, the Franel-Landau theorem established that the Riemann Hypothesis is equivalent to certain estimates on the discrepancy of Farey sequences. If rho_nu denotes the nu-th element of the Farey sequence of order N, the distribution of these points is nearly uniform. The source paper extends this classical framework by developing a generalized summation formula that links the values of a function h evaluated at Farey points to the Mertens function M(x).

The specific problem addressed in this analysis is the convergence of Riemann sums and their weighted averages. By representing the sum of a function over Farey points as a convolution involving the Mertens function and standard Riemann sums R_n(h), the paper provides a powerful tool for analyzing the error terms in the Prime Number Theorem. This approach is significant because it translates the analytical properties of the Riemann zeta function zeta(s) into the language of discrete summations and matrix transformation methods, such as the Toeplitz summation.

Mathematical Background

To understand the contributions of arXiv:hal-00353820v1, one must first define the Farey sequence of order x. The sequence consists of all irreducible fractions a/b such that 1 ≤ a ≤ b ≤ [x]. The number of elements in the sequence, denoted by Phi(x), is approximately (3/pi^2)x^2 as x approaches infinity. The Riemann sum of a function h over n points is defined as R_n(h) = (1/n) sum_{m=1 to n} h(m/n).

The source paper introduces a crucial identity that relates the sum of h over the Farey sequence to these Riemann sums: sum h(rho_nu) = sum n R_n(h) M([x]/n). This formula is derived using the properties of the Mobius function mu(d). Recall that the condition (a, b) = 1 (coprimality) can be expressed using the identity sum_{d|(a,b)} mu(d) = 1 if (a, b) = 1, and 0 otherwise. By substituting this into the sum over Farey points and rearranging the order of summation, the paper establishes that the Farey sum is essentially a weighted average of standard Riemann sums, where the weights are determined by the Mertens function.

The Mertens function M(x) = sum_{n≤x} mu(n) is central to the Riemann Hypothesis. RH is equivalent to the statement that M(x) = O(x^(1/2 + epsilon)) for any epsilon > 0. If this bound holds, the error term in the distribution of Farey points is significantly constrained. The paper also utilizes Bernoulli polynomials, specifically B_1(t) = t - 1/2, to refine the integral representation of the error term.

Technical Analysis of Equidistribution

The Mertens Transformation and Toeplitz Summation

The core technical achievement of the research is the transformation of the Farey sum into a form amenable to the Toeplitz limit theorem. The paper demonstrates that the average of h over Farey points converges to the integral of h as x approaches infinity, provided that the Riemann sums R_n(h) converge to the integral. The paper uses the identity sum n R_n(h) M([x]/n) to apply the Toeplitz lemma. It defines an array of reals a_{k,l} and shows that the conditions for the convergence of the sequence are satisfied. Specifically, the paper proves that the sum over (n/Phi(x)) |M(x/n)| is bounded. This bound ensures that the Farey points inherit the convergence properties of the standard Riemann sums.

Quadratic Discrepancy and GCD-Kernels

A striking result in arXiv:hal-00353820v1 is the formula for the variance of the Farey points: sum (rho_nu - nu/Phi(x))^2 = (1 / 12Phi(x)) { sum sum M(x/a) M(x/b) (a,b)^2/ab - 1 }. This identity provides a direct bridge to the Riemann Hypothesis. The double sum involves the Mertens function and the greatest common divisor (a, b). The term (a,b)^2/ab represents a kernel that weights the interactions between different scales of the Mertens function.

This structure suggests that the distribution of Farey sequences is not just a statistical property but a reflection of the arithmetic correlations encoded in the Mobius function. The derivation of this variance formula utilizes the properties of the L^2 norm of the discrepancy function, linking it to the spectral properties of the Farey sequence.

Novel Research Pathways

Pathway 1: Spectral Norms of the GCD-Kernel

One promising research direction involves a spectral analysis of the symmetric positive kernel K(a,b) = gcd(a,b)^2/(ab). The discrepancy identity shows that the quadratic equidistribution statistic is essentially a bilinear form. Under the assumption of RH, the vector M(X/a) should exhibit square-root cancellation. By studying K as an operator on weighted spaces, researchers could potentially prove that sharp bounds on this bilinear form are equivalent to the Riemann Hypothesis. This would turn a distributional problem into a spectral problem in functional analysis.

Pathway 2: Index-Localizing Test Functions

Another avenue is to develop a family of test functions h_n for which the Riemann sum R_m(h_n) is sharply peaked near m=n. If one can identify a function class for which the control of the discrepancy D_X(h) forces Mertens bounds close to RH, it would create a reversible implication between equidistribution speed and the critical line. This requires constructing functions with controlled derivatives and specific oscillatory structures that can probe the values of M(X/n) individually.

Computational Implementation

The following Wolfram Language code demonstrates the relationship between the Mertens function and the distribution of Farey points, verifying the core identity from the paper.

(* Section: Farey Sequence and Mertens Function Analysis *)
(* Purpose: This code computes the error term in Farey sums and compares it to the growth of the Mertens function *)

Module[{maxN, h, mertens, fareySum, riemannSumWeight, errorTerm, plotM, plotE},
  maxN = 80;
  h[t_] := t^2; (* Test function h(t) *)
  
  (* 1. Compute Mertens Function M(n) *)
  mertens = Accumulate[Table[MoebiusMu[k], {k, 1, maxN}]];
  
  (* 2. Generate Farey Sequence Sums *)
  fareySum[n_] := Module[{pts},
    pts = Select[Flatten[Table[a/b, {b, 1, n}, {a, 1, b}]], GCD[Numerator[#], Denominator[#]] == 1 &];
    pts = Union[pts];
    Total[h /@ pts]
  ];
  
  (* 3. Compute the Weighted Riemann Sum using the Identity *)
  (* Identity from arXiv:hal-00353820v1: Sum h(rho) = Sum n * R_n(h) * M(N/n) *)
  riemannSumWeight[N_] := Total[Table[
    n * (1/n * Total[Table[h[m/n], {m, 1, n}]]) * mertens[[Floor[N/n]]],
    {n, 1, N}
  ]];
  
  (* 4. Data Generation for Visualization *)
  errorTerm = Table[{n, Abs[fareySum[n] - riemannSumWeight[n]]}, {n, 1, maxN}];
  
  (* 5. Output Results *)
  Print["Average Difference (Verification): ", Mean[errorTerm[[All, 2]]]];
  ListLinePlot[errorTerm, PlotLabel -> "Identity Verification Error", AxesLabel -> {"N", "Error"}]
]

Conclusions

The analysis of arXiv:hal-00353820v1 reveals a deep and rigorous connection between the discrete world of Farey sequences and the analytic world of the Riemann zeta function. By expressing Farey sums as convolutions of the Mertens function and standard Riemann sums, the paper provides a unified framework for understanding the discrepancy of rational points. The variance formula presented is particularly significant, as it provides a concrete arithmetic identity that mirrors the requirements of the Riemann Hypothesis. The most promising avenue for further research lies in the spectral analysis of the gcd-kernel and the exploration of almost everywhere convergence in weighted Riemann sums. These paths offer the potential to bridge the gap between multiplicative number theory and dynamical systems.

References

• Source Paper: arXiv:hal-00353820v1
• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-function. Oxford University Press.
• Franel, J. (1924). Les suites de Farey et le probleme de nombres premiers. Nachr. Ges. Wiss. Gottingen.