Arithmetic Discrepancy and the Critical Line: Farey Sequences and the Mertens Function

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Introduction

The distribution of rational numbers organized into Farey sequences serves as a profound bridge between elementary arithmetic and the deep analytic properties of the Riemann zeta function. The source paper arXiv:hal-00353820 investigates the intricate relationship between the local properties of these fractions and the global behavior of the Mertens function. This connection is fundamentally linked to the Riemann Hypothesis (RH), which posits that the non-trivial zeros of the zeta function ζ(s) lie on the critical line Re(s) = 1/2.

Historically, the Franel-Landau theorem established the initial equivalence between the uniform distribution of Farey sequences and the growth rate of the Mertens function. This article synthesizes identities from the source to demonstrate how the variance of Farey discrepancies mirrors the distribution of prime numbers and how modern functional analysis can be used to probe the critical strip.

Mathematical Background

The Farey sequence of order N, denoted as F_N, consists of irreducible fractions in the interval [0, 1] with denominators not exceeding N, arranged in increasing order. The number of elements in this sequence, Φ(N), grows asymptotically as 3N²/π². A central object of study in arXiv:hal-00353820 is the sum of a test function h over the Farey points ρ_ν.

The paper establishes a pivotal identity: the sum of h(ρ_ν) for ν from 1 to Φ(N) is equal to the sum over n of n R_n(h) M([N]/n), where M(y) is the Mertens function and R_n(h) is a weighted discrete average. This identity is remarkable because it isolates the number-theoretic difficulty into the fluctuations of the Mobius function, which is the summatory component of the Mertens function. The growth of M(x) is equivalent to the Riemann Hypothesis if it remains bounded by O(x^{1/2 + ε}) for any ε > 0.

Spectral Properties and Zero Distribution

The analysis in arXiv:hal-00353820 moves beyond simple summation identities to consider the spectral nature of arithmetic averages. An identity in the source connects the integral of squared remainder terms R_n² to the divisor function d(k). This link is significant because the divisor function appears in the square of the zeta function, ζ(s)². The convergence of these sums is therefore inextricably linked to the growth of ζ(s) on the critical line.

Furthermore, the paper discusses the convergence of an operator A_N, which represents the average of the remainders. This ergodic result suggests that the Riemann Hypothesis can be interpreted as a statement about the rate of decay in the correlations of the underlying dynamical system. If the correlations decay at a rate consistent with square-root cancellation, the central limit theorem for the Mobius function holds, implying the validity of RH.

Moment Estimates and Growth Rates

One of the most technically significant results in arXiv:hal-00353820 is the explicit formula for the variance of Farey discrepancy. The paper shows that the sum of (ρ_ν - ν/Φ(x))² is proportional to a quadratic form involving the Mertens function and the greatest common divisor (a,b). Specifically, the variance is expressed as a double sum of M(x/a)M(x/b) weighted by (a,b)²/ab.

Geometric Interpretation: This formula acts as a direct bridge, transforming RH into a statement about the L² geometry of rational points.
Scale Decomposition: The presence of the term (a,b)²/ab suggests a multi-scale decomposition where RH is equivalent to specific eigenvalue bounds on the matrix of arithmetic correlations.
Convergence Criteria: The almost-everywhere convergence of these arithmetic averages provides a rigorous framework for asserting that the fluctuations of the Mobius function are stable across different scales.

Novel Research Pathways

Spectral Analysis of Multiplicative Kernels

A promising direction involves the diagonalization of the kernel K(a,b) = gcd(a,b)²/(ab). By applying Mellin transforms, researchers can convert the quadratic discrepancy into integrals involving ζ(s) and 1/ζ(s). A successful Plancherel-type identity would localize contributions near the critical line, revealing which modes obstruct optimal discrepancy bounds.

Generalized Farey Statistics and L-functions

The techniques in arXiv:hal-00353820 can be extended to study generalized Farey sequences associated with algebraic number fields. This would relate the distribution of rational points in number fields to the zeros of Dedekind zeta functions and L-functions, potentially offering new tools for investigating the Generalized Riemann Hypothesis (GRH).

Computational Implementation

Wolfram Language

(* Section: Farey Discrepancy and Zeta Zero Correlation *)
(* Purpose: Compute the Farey discrepancy and visualize its relationship to the Mertens function and Zeta zeros *)

Module[{nMax = 40, fareyFractions, phiN, discrepancy, mValues, zeros, rescaledD},
  (* Generate reduced Farey fractions for order nMax *)
  fareyFractions = Sort[DeleteDuplicates[Flatten[Table[a/b, {b, 1, nMax}, {a, 0, b}]]]];
  phiN = Length[fareyFractions];

  (* Calculate L2 discrepancy: sum of (rho_nu - nu/phiN)^2 *)
  discrepancy = Total[Table[(fareyFractions[[i]] - (i - 1)/(phiN - 1))^2, {i, 1, phiN}]];
  rescaledD = nMax * discrepancy;

  (* Mertens function values for comparison with square root growth *)
  mValues = Table[{x, Total[MoebiusMu[Range[x]]]}, {x, 1, nMax}];

  (* Obtain first few non-trivial zeros of the Riemann Zeta function to show critical line *)
  zeros = Table[ZetaZero[k], {k, 1, 15}];

  (* Visualizing the results *)
  Print["Farey Order N: ", nMax];
  Print["Sequence Length: ", phiN];
  Print["Total Discrepancy D2: ", discrepancy];
  Print["Rescaled Discrepancy N*D2: ", rescaledD];

  (* Plotting the fluctuations of the Mertens Function *)
  Print[ListLinePlot[mValues,
    PlotLabel -> "Mertens Function Fluctuations",
    AxesLabel -> {"x", "M(x)"},
    PlotStyle -> Red]];

  (* Plotting the Imaginary parts of Zeta zeros *)
  Print[ListPlot[Table[{1/2, Im[z]}, {z, zeros}],
    PlotLabel -> "Zeta Zeros on the Critical Line",
    AxesLabel -> {"Re", "Im"},
    PlotRange -> {{0, 1}, All}]];

  (* Return summary statistics *)
  {nMax, phiN, discrepancy}
]

Conclusions

The investigations in arXiv:hal-00353820 demonstrate that Farey equidistribution is a rigorous reflection of arithmetic structure. The explicit link between discrepancy variance and the Mertens function transforms a problem of complex analysis into one of discrete geometry. The most promising avenue for further research lies in the spectral analysis of the associated transfer operators. Establishing a lower bound on the spectral gap would correspond directly to the width of the critical strip, potentially leading to a resolution of the Riemann Hypothesis through the lens of rational approximation.

References

arXiv:hal-00353820: Sur la suite de Farey et l'hypothese de Riemann.
Franel, J. (1924). Sur la serie des fractions de Farey et l'hypothese de Riemann. Vierteljahresschr. Naturforsch. Ges. Zurich, 69, 198-201.
Landau, E. (1924). Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel. Gottinger Nachrichten, 269-277.