Spectral Rigidity and Hyperbolic Mertens Increments in the Nyman-Beurling Criterion

Introduction

The Nyman-Beurling approach to the Riemann Hypothesis (RH) represents one of the most elegant intersections of functional analysis and number theory. It recasts the distribution of the nontrivial zeros of the Riemann zeta function, ζ(s), into a question of the density of specific function subspaces within a Hilbert space. Specifically, RH is equivalent to the statement that the characteristic function of the interval (0, 1] lies in the closure of a subspace generated by dilations of the fractional part function. The research paper arXiv:hal-01950436 advances this program by developing a discrete realization involving step functions constant on unit intervals.

This discrete framework, often associated with the Baez-Duarte refinement, translates the analytic problem into the study of sequences and Dirichlet series. By examining the properties of functions that are constant on each interval [j, j+1[, the paper provides a bridge between the discrete summatory behavior of arithmetic functions and the continuous analytic continuation of ζ(s). The central contribution of the source involves establishing precise norm estimates for approximating sequences and demonstrating how the existence of zeros off the critical line would impose rigid vanishing constraints on the associated transforms.

Mathematical Background

The foundation of the analysis in arXiv:hal-01950436 is a Hilbert space H of functions defined on the positive integers, or more accurately, functions on the real line that are step-wise constant. Let φ be a function constant on each interval [j, j+1[ for j in N. The paper defines a subspace D where functions vanish on [0, 1[. A primary object of study is the sequence of basis functions ε_k and the expansion of the target vector e₁.

The inner product structure in this space is intimately tied to the harmonic weight 1/n. For example, the expansion of the vector e₁′ is given by:

e₁′ = sum_{k ≥ 1} <e₁, ε_k> ε_k = sum_{k ≥ 1} sqrt(k(k+1))(ln(1+1/k) - 1/(k+1)) ε_k.

The coefficients here decay at a rate of approximately 1/(2k), placing the expansion on the boundary of square-summability. This logarithmic behavior is characteristic of structures where the Mellin transform interacts with the zeta function. Furthermore, the paper associates each function f in D with a Dirichlet series F(s) = sum f(n)n^-s. According to Proposition 8, this series converges absolutely for σ > 3/2 and admits a meromorphic continuation to the half-plane σ > 1/2. The behavior of the quotient F(s)/ζ(s) in this region is the focal point for proving or refuting the density required by the Nyman-Beurling criterion.

Main Technical Analysis

Hyperbolic Möbius Increments and Norm Bounds

A central result in arXiv:hal-01950436 is the derivation of a bound for the squared norm of the approximation sequence S_K. This sequence is constructed to approximate the characteristic function, and its norm represents the "distance" to the target. The paper establishes that:

||S_K||² = sum_{d ≤ K-1} d(d+1) ( m(K/d)/d - m(K/(d+1))/(d+1) )² + 1 + 1/K.

Using standard inequalities, this is bounded as:

||S_K||² ≪ 1 + sum_{d ≤ K-1} ( m(K/d) - m(K/(d+1)) )².

The term (m(K/d) - m(K/(d+1))) represents the increment of the Mertens function m(x) over the interval (K/(d+1), K/d]. These are referred to as hyperbolic intervals. The significance of this bound is profound: it links the Hilbert space distance directly to the second moment of Möbius sums over a specific hyperbolic partition of the interval [1, K]. Under the Riemann Hypothesis, one expects square-root cancellation in these increments, which would constrain the growth of ||S_K||.

Dirichlet Series and Analytic Continuation

The paper moves from norm estimates to the analytic properties of the associated transforms. Proposition 9 states that for f in D, the Dirichlet series F(s)/ζ(s) is itself a Dirichlet series in a suitable half-plane. The existence of a meromorphic continuation of F(s) to σ > 1/2 implies that if ζ(s) has a zero ρ with Re(ρ) > 1/2, then F(s) must also vanish at ρ to prevent a pole in the quotient.

This lead to a "rigidity" requirement: the Mellin transform of the sequence must vanish at every zero of the zeta function in the critical strip. The paper argues that if the space D is sufficiently rich, the only function satisfying these vanishing conditions across all potential off-line zeros would be the zero function itself. Thus, the completeness of the subspace is directly obstructed by any zeros of ζ(s) that lie to the right of the critical line Re(s) = 1/2.

Spectral Properties and Orthonormal Systems

The analysis of the coefficients <S_K, ε_d> reveals that for any fixed d, the inner product tends to zero as K approaches infinity. This indicates that the approximation sequence S_K converges weakly to zero. In the context of Hilbert space operators, this weak convergence is a necessary step toward strong convergence, which would imply the truth of RH. The paper specifically calculates the supremum of certain inner products, such as sup |<e₁, f_n>| = ln(4) - 1, providing concrete geometric constants that bound the approximation capability of the system.

Novel Research Pathways

Based on the findings of arXiv:hal-01950436, several research directions emerge:

• Quantitative Hyperbolic Moments: Investigating whether the growth condition sum (m(K/d) - m(K/(d+1)))² = O(K^ε) is strictly equivalent to RH. This would involve using Mellin inversion to relate the sum of increments to an integral of |1/ζ(1/2+it)|².
• Spectral Gap of the Möbius Operator: Defining a "Möbius transform" operator T on the step-function space and studying its spectral gap. A positive lower bound on the spectral gap would imply limits on the density of zeta zeros near the critical line.
• Weighted Nyman-Beurling Spaces: Introducing a weight θ^α into the L² norm to probe zero-free regions σ > (1+α)/2. This could allow for a continuous deformation of the Nyman-Beurling criterion.

Computational Implementation

The following Wolfram Language implementation explores the hyperbolic Mertens-increment bound discussed in the paper. It calculates the bound surrogate B(K) and compares it against the distribution of zeta zeros.

(* Section: Hyperbolic Mertens-increment bound *)
(* Purpose: Numerically explore the bound surrogate B(K) from hal-01950436 *)

ClearAll[mertensSum, boundB, kValues, results];

(* summatory Moebius function m(x) *)
mertensSum[n_Integer] := Total[MoebiusMu[Range[n]]];
mertensSum[0] = 0;

(* Calculate B(K) = 1 + Sum_{d=1 to K-1} (m(K/d) - m(K/(d+1)))^2 *)
boundB[K_Integer] := Module[{sumVal = 1, d, m1, m2},
  Do[
    m1 = mertensSum[Floor[K/d]];
    m2 = mertensSum[Floor[K/(d + 1)]];
    sumVal += (m1 - m2)^2,
    {d, 1, K - 1}
  ];
  sumVal
];

(* Generate data for K from 10 to 500 *)
kValues = Range[10, 500, 10];
results = Table[{k, boundB[k]}, {k, kValues}];

(* Visualization *)
ListLinePlot[results, 
  PlotRange -> All, 
  AxesLabel -> {"K", "B(K)"}, 
  PlotLabel -> "Hyperbolic Mertens Increment Second Moment",
  PlotStyle -> Blue]

(* Reference: First 5 Zeta Zeros *)
Table[N[Im[ZetaZero[n]]], {n, 1, 5}]

Conclusions

The framework established in arXiv:hal-01950436 provides a rigorous path for connecting the discrete summatory properties of the Möbius function to the analytic continuation of the Riemann zeta function. By bounding the approximation norm by hyperbolic increments of the Mertens function, the paper identifies the specific scale at which arithmetic cancellations must occur to support the Riemann Hypothesis. The most promising avenue for future work lies in the spectral analysis of the operators acting on these step-function spaces, particularly in determining if the observed weak convergence of the approximation sequence can be strengthened to the strong convergence required for a definitive proof. Specific next steps include the refinement of the upper bounds using smoothed Mertens functions to enable more precise Mellin-Plancherel identities.

References

• Source Paper: arXiv:hal-01950436
• L. Báez-Duarte, "A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis," 2003.
• B. Nyman, "On the one-dimensional translation group and semi-group in certain function spaces," 1950.
• A. Beurling, "A closure problem related to the Riemann zeta-function," 1955.