Short Interval Moment Asymptotics and the Geometric Rigidity of Zeta Zeros

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Introduction

The Riemann Hypothesis remains the most significant challenge in analytic number theory, asserting that the non-trivial zeros of the Riemann zeta function, ζ(s), are restricted to the critical line where the real part of s is 1/2. While global asymptotics for the moments of the zeta function have been studied for over a century, the focus has recently shifted toward short-interval mean values. These local estimates, as explored in the source paper arXiv:hal-01112342, provide a higher resolution view of the function's growth and its relationship with the distribution of nearby zeros.

The study of short intervals—intervals of length H that are significantly smaller than the height T—is intrinsically linked to the Lindelöf Hypothesis and the GUE (Gaussian Unitary Ensemble) conjecture. By examining how the function behaves in windows as small as T to the power of θ, researchers can probe the fine-scale fluctuations that global averages often smooth out. The paper arXiv:hal-01112342 establishes rigorous bounds and explicit formulas that connect these moments to the oscillatory behavior of the zeta function error terms.

This technical analysis synthesizes the findings of the source paper to demonstrate how moment estimates on the critical line serve as proxies for zero-density results. We examine the role of smoothing kernels, the Atkinson formula, and the fourth-moment error term in constraining the analytic landscape of the critical strip. By bridging these classical analytic methods with modern computational insights, we identify the most promising pathways for future research into the vertical distribution of zeros.

Mathematical Background

The primary objects of study are the 2k-th moments of the zeta function on the critical line, defined by the integral of |ζ(1/2 + it)| raised to the power of 2k. In the context of arXiv:hal-01112342, the focus is on the increment of these moments over a short interval [T - H, T + H]. For k=1 (the mean square) and k=2 (the fourth moment), the behavior of these integrals is governed by a main term and a fluctuating error term, denoted as E(T) and E₂(T) respectively.

A central pillar of this analysis is the Atkinson Formula. This result provides an explicit representation of E(T) as a sum of two oscillatory components, involving the divisor function d(n) and a phase function f(T, n). The phase function represents the fundamental frequency of the zeta function's fluctuations and is given by the expansion involving arsinh and square-root terms. Specifically, for small n, the phase behaves as 2 times the square root of 2πnT.

The paper also utilizes the generalized divisor function, d_k(n), which appears in the Dirichlet series expansion of ζ(s)^k. The behavior of these coefficients, combined with smoothing techniques using kernels like the double exponential exp(-c₁ exp(|v-t|/3)), allows for the isolation of local growth. This methodology is essential for proving that the moments do not vanish or grow uncontrollably in short intervals, a property that is highly sensitive to the presence of zeros off the critical line.

Main Technical Analysis

Smoothing Kernels and Short-Interval Inequalities

A critical contribution of arXiv:hal-01112342 is the use of super-exponentially decaying kernels to localize the zeta function. The paper employs an integral inequality where the length H is bounded by a weighted integral of the zeta function. By choosing a kernel that concentrates mass within a distance of order 1 from the center of the interval, the analysis converts pointwise growth questions into mean-value problems. This is particularly effective for establishing lower bounds on moments, showing that the zeta function cannot be "too small" for too long—a direct consequence of the rigidity imposed by the zeros on the critical line.

Atkinson Phases and Spectral Oscillations

The analysis of the error term E(T) relies on the precise expansion of the phase function f(T, n). The source paper provides a high-order expansion: f(T, n) = -π/4 + 2 sqrt(2πnT) + (1/6) sqrt(2π³) n^3/2 T^-1/2 + higher order terms. This square-root dependence on the height T is a signature of the spectral properties of the zeta function. In short intervals, the difference between phases at T+H and T-H determines the cancellation within the sum. If the zeros are distributed according to GUE statistics, these oscillations should lead to the observed square-root cancellation in the error term.

The Fourth Moment Error Term E₂(T)

The fourth moment is significantly more complex than the mean square. The paper record bounds for E₂(T) as being O(T^2/3 log^C T) and Ω(T^1/2). The gap between these exponents represents one of the major frontiers in analytic number theory. The source paper investigates the integrated error, showing that the average of E₂(T) over T is O(T^3/2). This implies that while the error can reach large peaks, its "energy" is constrained, suggesting a level of regularity in the fourth moment that is consistent with the Riemann Hypothesis.

Novel Research Pathways

Pathway 1: Incremental Rigidity and the Lindelöf Hypothesis

Future research should focus on bounding the increments E₂(T+H) - E₂(T) for very small H. If one can prove that these increments are small for H = T^ε, it would imply that the fourth moment is locally stable. This pathway involves applying van der Corput's method for exponential sums to the Atkinson-like representation of the fourth moment. The expected outcome is a refinement of the subconvexity bound for the zeta function, moving the exponent closer to 0, as predicted by the Lindelöf Hypothesis.

Pathway 2: Measure-Theoretic Tail Bounds and Large Values

The source paper derives a bound for the measure μ(T, H, V) of points where log |ζ| exceeds V, showing it decays as exp(-θ V log V). A promising direction is to investigate the transition from this tail behavior to the Gaussian behavior predicted by Selberg's Central Limit Theorem. By refining the Dirichlet polynomial approximations used in arXiv:hal-01112342, researchers could establish a uniform distributional model for the zeta function in short intervals, which would provide a quantitative measure of how zeros "repel" the function's values away from the origin.

Computational Implementation

The following Wolfram Language implementation explores the short-interval behavior of the error term E(T) by comparing the numerical integral of the zeta function against the theoretical main term derived from the mean value theorem.

(* Section: Short-Interval Error Analysis *)
(* Purpose: Numerically evaluate the error term E(T) near height T0 *)

Module[{T0 = 5000, H = 10, precision = 20, steps = 50, mainTerm, data},
  
  (* Define the main term for the mean square: T(log(T/2pi) + 2gamma - 1) *)
  mainTerm[t_] := t * (Log[t / (2 * Pi)] + 2 * EulerGamma - 1);

  (* Calculate the integral of |zeta(1/2+it)|^2 in short windows *)
  data = Table[
    Module[{currentT = T0 + i, integral, error},
      integral = NIntegrate[
        Abs[Zeta[1/2 + I * t]]^2, 
        {t, 0, currentT}, 
        PrecisionGoal -> 8, 
        WorkingPrecision -> precision
      ];
      error = integral - mainTerm[currentT];
      {currentT, error}
    ], 
    {i, 0, steps, 1}
  ];

  (* Visualize the oscillations of E(T) *)
  ListLinePlot[data, 
    PlotRange -> All, 
    AxesLabel -> {"T", "E(T)"}, 
    PlotLabel -> "Oscillations of the Mean Square Error Term",
    PlotTheme -> "Scientific",
    Filling -> Axis
  ]
]

Conclusions

The analysis of the Riemann zeta function in short intervals, as presented in arXiv:hal-01112342, reveals a profound structural coherence that global averages fail to capture. The establishment of sharp lower bounds for moments and the detailed expansion of oscillatory phases provide a robust framework for testing the limits of the zeta function's growth. The most promising avenue for future investigation lies in the spectral analysis of the error terms E(T) and E₂(T), which act as the primary interface between the arithmetic properties of the zeta function and the geometric distribution of its zeros.

Ultimately, the rigidity observed in short-interval moments suggests that the fluctuations of the zeta function are far from random; they are governed by an underlying symmetry that is consistent with the Riemann Hypothesis. Continued refinement of these methods, particularly in the regime of ultra-short intervals, will be essential for narrowing the search for zeros and proving the subconvexity bounds necessary for the next generation of prime number theorems.

References

arXiv:hal-01112342: Mean values results for the Riemann zeta-function in short intervals.
Atkinson, F. V. (1949). The mean-value of the Riemann zeta function. Acta Mathematica, 81, 353-376.
Ivić, A. (2003). The Riemann Zeta-Function: Theory and Applications. Dover Publications.