Asymptotic LCM Distributions and the Probabilistic Structure of the Riemann Zeta Function

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Introduction

The Riemann Hypothesis remains the most profound challenge in analytic number theory, asserting that the non-trivial zeros of the Riemann zeta function ζ(s) are restricted to the critical line Re(s) = 1/2. While traditionally approached through complex analysis, the paper arXiv:hal-01984389 provides a sophisticated probabilistic entry point by investigating the asymptotic behavior of the least common multiple (LCM) of several random integers. By modeling the p-adic valuations of integers as independent geometric random variables, the source paper constructs a framework where the distribution of multiplicative functions is governed by infinite products over primes.

The significance of this approach lies in the transition from finite integer samples to infinite Euler products. The paper demonstrates that the LCM of k random integers, when properly normalized, converges to a distribution that can be decomposed into independent local factors at each prime. However, the precision of this convergence—the speed at which the finite-n model approaches the infinite-prime limit—is intrinsically tied to the distribution of prime numbers and, consequently, the zeros of the zeta function. This article establishes the formal connections between these probabilistic limits and the analytic properties of the critical strip.

Mathematical Background

The core mathematical object in arXiv:hal-01984389 is the random variable X⁽ⁿ⁾, distributed uniformly on the set {1, 2, ..., n}. For a prime p, the p-adic valuation λ_p(m) is the highest power of p dividing m. The paper shows that for any finite set of primes, the joint distribution of valuations converges as n tends toward infinity:

Geometric Convergence: P(λ_p(X⁽ⁿ⁾) ≥ c) = (1/n) floor(n / p^c) which approaches p^-c.
Independence: In the limit, the valuations for different primes behave as independent geometric random variables G(p) with parameter 1 - 1/p.
LCM Structure: The valuation of the LCM of k independent samples is the maximum of their individual valuations: λ_p(L_n(k)) = max(λ_p(X₁⁽ⁿ⁾), ..., λ_p(X_k⁽ⁿ⁾)).

The limiting distribution of the normalized LCM is expressed as an infinite product over primes. This product is essentially a random Euler product. Since the Riemann zeta function is defined by the deterministic Euler product ζ(s) = product (1 - p^-s)^-1, the statistical fluctuations of L_n(k) provide a direct probabilistic analogue to the analytic properties of ζ(s).

Main Technical Analysis

Spectral Properties of Valuation Generating Functions

The source paper derives explicit generating functions for the maximum valuation across k samples. For k = 2, the expectation takes the form:

E[t^Z₂(p)] = (1 - 1/p²) / (1 - t/p²)

This rational function of p and t serves as a local Euler factor. When we consider the product over all primes, we obtain a function whose analytic continuation is determined by ζ(2) and its related values. For higher k, such as k = 3, the paper provides a more complex structure involving terms like (1 - t/p²) and (1 - t²/p³) in the denominator. The poles of these local factors, when aggregated, define the singularity structure of the global distribution, echoing how the zeros of ζ(s) determine the distribution of primes.

Truncation Analysis and Prime-Counting Fluctuations

A critical technique in arXiv:hal-01984389 is the truncation of the prime set at a value m, followed by a double limit (n to infinity, then m to infinity). The paper proves that the contribution of "large" primes (p > m) vanishes in probability. Specifically, it analyzes the range (m, sqrt(n)] and (sqrt(n), n].

The bound on the sum over p > m of the probability that a prime divides more than one sample is given by k² sum(p^-2). While this sum converges unconditionally, a sharper effective bound on the convergence rate of the LCM distribution requires controlling the sum of 1/p. The error term in the sum of 1/p up to x is known to be O(exp(-c sqrt(log x))) unconditionally, but under the Riemann Hypothesis, it improves to O(x^-1/2 log x). Thus, the "probabilistic distance" between the random LCM and its limit is a zero-sensitive quantity.

Moment Estimates and the Critical Strip

The paper establishes that the normalized moments of the LCM converge to constants C_r,k. These constants are infinite products of local expectations. If we evaluate these products as functions of a complex parameter s, we find that they are convergent for Re(s) > 1. The behavior of these moments as s approaches the critical line Re(s) = 1/2 is where the connection to the Riemann Hypothesis is most acute. The variance of the logarithm of the LCM grows logarithmically with n, a scaling behavior that matches the variance of the log-zeta function on the critical line, as described by Selberg's Central Limit Theorem.

Novel Research Pathways

Pathway 1: Effective Convergence Rates and Zeta Zeros

We propose investigating the explicit rate of convergence for the distribution of log(L_n(k)). By applying the Berry-Esseen theorem to the sum of independent-like valuations, one can attempt to bound the Kolmogorov distance between the finite-n distribution and the geometric limit. The optimal error term is expected to be a function of the supremum of the real parts of the zeta zeros. Proving a rate of O(n^{-1/2 + epsilon}) would provide a new probabilistic criterion for the Riemann Hypothesis.

Pathway 2: Expected Euler Products and Analytic Continuation

Another direction involves constructing the "Expected Euler Product" function F(s) = product E[p^{-s G(p)}]. Using the generating functions from the source paper, one can show that F(s) can be factored into a product of zeta functions (e.g., ζ(s+1), ζ(2s+1)). Research should focus on whether the probabilistic independence of G(p) can be used to prove the non-vanishing of F(s) in regions of the critical strip, thereby providing a back-door to zero-free region results.

Computational Implementation

The following code simulates the LCM valuation statistics described in arXiv:hal-01984389 and compares them to the theoretical Euler product limits, while also visualizing the zeta zeros that govern the error terms.

(* Section: Random LCM Statistics and Zeta Zeros *)
(* Purpose: Compare simulated LCM valuations with Euler product theoretical limits *)

ClearAll[lcmValuation, theoreticalLimit, zetaZeroPlot];

(* Simulate max p-adic valuation for k random integers *)
simulateLCM[n_, k_, p_, trials_] := Module[{vals},
  vals = Table[
    Max[Table[IntegerExponent[RandomInteger[{1, n}], p], {k}]],
    {trials}
  ];
  Counts[vals] / N[trials]
];

(* Theoretical probability P(max valuation = m) from hal-01984389 *)
theoreticalLimit[p_, k_, m_] := (1 - p^-(m + 1))^k - (1 - p^-m)^k;

(* Parameters for p=2, k=3 *)
nVal = 10^5; kVal = 3; pVal = 2; trials = 5000;
simData = simulateLCM[nVal, kVal, pVal, trials];
theoryData = Table[theoreticalLimit[pVal, kVal, m], {m, 0, 5}];

(* Visualization of the distribution *)
distPlot = BarChart[{KeySort[simData], theoryData}, 
  ChartLabels -> {"Simulation", "Theory"}, 
  PlotLabel -> "LCM Valuation Distribution (p=2, k=3)"];

(* Visualization of Zeta Zeros on the Critical Line *)
zetaPlot = Plot[Abs[Zeta[1/2 + I t]], {t, 0, 50}, 
  PlotLabel -> "Zeta Magnitude on Critical Line", 
  AxesLabel -> {"t", "|zeta(1/2+it)|"}];

Print[distPlot];
Print[zetaPlot];
Print["First 5 Zeta Zeros:", Table[N[Im[ZetaZero[j]]], {j, 1, 5}]];

Conclusions

The analysis of random least common multiples in arXiv:hal-01984389 reveals a profound structural similarity between probabilistic valuation models and the analytic theory of the Riemann zeta function. The convergence of normalized LCMs to infinite Euler products provides a framework where prime distribution fluctuations are translated into statistical variances. The most promising avenue for further research lies in the quantification of convergence rates, as the transition from finite samples to the geometric limit is governed by the same error terms that define the Riemann Hypothesis. Future work should focus on extending these probabilistic models to Dirichlet L-functions to explore generalized versions of the hypothesis.

References

Primary Source: arXiv:hal-01984389
Tenenbaum, G. (2015). Introduction to Analytic and Probabilistic Number Theory.
Selberg, A. (1992). Old and new conjectures and results about a class of Dirichlet series.