The Probability of Prime Maxima: How Random Least Common Multiples Mirror the Zeta Function

Download Full Article

This article is available as a downloadable PDF with complete code listings and syntax highlighting.

Introduction

The study of the least common multiple (LCM) of a set of random integers represents a profound intersection between probabilistic number theory and analytic arithmetic. The source paper, arXiv:hal-01984389v1, provides a rigorous framework for understanding the limit theorems associated with the LCM of a sample of size k drawn uniformly from the set {1, 2, ..., n}. While the paper focuses on the convergence in distribution of the normalized LCM and its moments, the underlying structures—specifically the prime power valuations and the reliance on the distribution of prime numbers—point directly toward the Riemann Hypothesis.

The Riemann Hypothesis (RH) posits that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. This is intrinsically linked to the error term in the Prime Number Theorem. Since the LCM of a set of integers is determined by the maximum power of each prime dividing at least one member of the set, the growth and fluctuations of the LCM function are sensitive to the density and distribution of primes. The work in arXiv:hal-01984389v1 generalizes the classical understanding of the LCM of the first n integers, denoted as L_n, by introducing a stochastic dimension that allows for the application of probabilistic limit theorems to classical number-theoretic problems.

Mathematical Background

In arXiv:hal-01984389v1, the primary object of study is the random variable L_n(k) = lcm(X₁, X₂, ..., X_k), where X_j are independent and identically distributed (i.i.d.) random variables following a uniform distribution on {1, ..., n}. To analyze L_n(k), the authors utilize the p-adic valuation λ_p(m), which denotes the exponent of the highest power of the prime p that divides m.

The LCM can be expressed as a product over the set of all primes P: L_n(k) = ∏ p^{max(λ_p(X₁), ..., λ_p(X_k))}. The paper establishes that as n approaches infinity, the vector of valuations converges in distribution to a sequence of independent geometric random variables G(p), where the probability P(G(p) ≥ c) = p^-c. This leads to the definition of a limit random variable X_f,∞,k, representing an infinite random Euler product.

This structure is reminiscent of the Euler product for the Riemann zeta function ζ(s) = ∏ (1 - p^-s)^-1. In the context of arXiv:hal-01984389v1, the moments of the normalized LCM involve sums over prime powers that are essentially Dirichlet series. The convergence of these series and the behavior of the tail probabilities are governed by the same analytic properties that determine the zero-free region of ζ(s).

Main Technical Analysis

Spectral Properties and Zero Distribution

A primary connection between arXiv:hal-01984389v1 and the Riemann Hypothesis lies in the structure of the local Euler factors derived from the probability generating functions. The paper establishes that for a fixed prime p, the random valuation λ_p(X⁽ⁿ⁾) converges to a geometric variable G(p). The generating function for the maximum of k such variables, Z_k(p), provides the local building blocks for a random Euler product.

As shown in the technical derivations, for k=2, the expectation E[t^Z₂(p)] = (1 - 1/p²) / (1 - t/p²). If one interprets t as a complex variable p^1-s, this expression takes the form of a ratio of zeta factors. The zeros and poles of these local factors, when aggregated across all primes, mirror the analytical structure of the zeta function near the critical strip. The presence of the term 1/p² ensures the absolute convergence of the variance, a property that is essential for maintaining the stability of the distribution on the critical line.

Sieve Bounds and Prime Density

The paper provides sophisticated bounds for the tail of the sum of random variables B_p,r. This analysis is partitioned into primes p ≤ m and p > m. The probability that the sum of these variables exceeds a threshold ε is shown to vanish as m and n go to infinity. This is essentially a sieve-theoretic argument that relies on the density of primes.

If the Riemann Hypothesis is true, the number of primes in any interval is highly predictable, with an error term of O(x^1/2 log x). The precision of the probability estimates in arXiv:hal-01984389v1 depends on this density. Any significant deviation from the expected prime distribution (such as large clusters or gaps) would disrupt the convergence rates of the LCM moments. Thus, the stability of the limit X_f,∞,k is a probabilistic reflection of the regularity of prime distribution necessitated by RH.

Moment Estimates and Growth Rates

The source paper investigates the convergence of moments E[(L_n(k) / n^k)^r] to a constant (r+1)^-k C_r,k. This convergence is non-trivial and depends on the behavior of the LCM for large primes. The paper demonstrates that the sum over primes of the probability P(∑ G_j(p) ≥ 2) is bounded by k² ∑ 1/p², which is finite.

This finiteness is crucial; it implies that for large primes, the probability of multiple integers in a sample sharing a high power of p is negligible. In analytic terms, this corresponds to the convergence of the Dirichlet series associated with the LCM. The growth rates of these moments provide a testing ground for the Lindelof Hypothesis, which is a consequence of RH and deals with the growth of the zeta function on the critical line.

Novel Research Pathways

Multiple Dirichlet Series and the LCM Zeta Function

One promising direction is to define a deterministic multiple Dirichlet series D_k(s) = ∑ lcm(n₁, ..., n_k)^-s. Using the combinatorial identities from arXiv:hal-01984389v1, such as the coefficient extraction for maximum valuations, one can factor D_k(s) into a product of shifted zeta functions and a remainder term H(s). The analytic continuation of H(s) beyond the line Re(s) = 1 would provide new information about the distribution of zeta zeros.

Mod-Gaussian Limits for Log-LCM Fluctuations

A second pathway involves studying the fluctuations of log(L_n(k)). Selberg's Central Limit Theorem states that log ζ(1/2 + it) behaves like a complex Gaussian. By applying the cumulant expansion techniques suggested by the paper's generating functions, researchers can investigate whether the normalized fluctuations of the log-LCM also converge to a Gaussian distribution at a rate sensitive to the heights of zeta zeros.

Discrepancy and the Square-root Barrier

The discrepancy between the joint law of valuations and the independent geometric model can be quantified. If this discrepancy can be bounded uniformly for primes up to n^θ, it would imply cancellations in Mertens-type sums. Establishing such bounds for θ ≥ 1/2 would be equivalent to proving results of the strength of the Riemann Hypothesis.

Computational Implementation

(* Section: Probabilistic LCM vs Zeta Distribution *)
(* Purpose: Simulate random LCM valuations and compare to Zeta factor logic *)

Module[{n = 10^5, k = 3, samples = 500, p = 2, valuations, zetaApprox},
  
  (* Simulate k-tuple LCM valuations for a fixed prime p *)
  valuations = Table[
    Max[Table[IntegerExponent[RandomInteger[{1, n}], p], {k}]],
    {samples}
  ];
  
  (* Theoretical Generating Function for k=2 from hal-01984389v1: (1-1/p^2)/(1-t/p^2) *)
  (* We can extend this to k=3 and compare empirical mean valuation *)
  Print["Empirical Mean Max Valuation at p=2: ", Mean[N[valuations]]];
  
  (* Plotting the distribution of valuations vs the geometric limit *)
  Print[Histogram[valuations, {0, Max[valuations] + 1, 1}, "Probability", 
    PlotLabel -> "Valuation Maxima Distribution (p=2)", 
    ChartStyle -> "Pastel"]];

  (* Visualize Zeta on the critical line for comparison *)
  zetaApprox = Plot[Abs[Zeta[1/2 + I t]], {t, 0, 40}, 
    PlotRange -> All, 
    Filling -> Axis, 
    PlotLabel -> "Zeta Function Amplitude on Critical Line",
    Frame -> True];
    
  Print[zetaApprox];
  
  (* Compute the first few Zeta zeros to show the distribution density *)
  Print["First 5 Zeta Zero Imaginaries: ", Table[Im[ZetaZero[j]], {j, 1, 5}]];
]

Conclusions

The analysis of random least common multiples provides a robust probabilistic mirror to the analytic properties of the Riemann zeta function. By establishing limit theorems for prime valuations and their maxima, arXiv:hal-01984389v1 offers a framework where the global properties of integers are decomposed into local p-adic components. The most promising avenue for further research lies in the study of the error terms in these limit theorems, as the transition from discrete samples to continuous limits involves prime sums where the Riemann Hypothesis is the governing factor. Future work should focus on deriving an explicit formula for the random LCM, expressing its deviations as a sum over the non-trivial zeros of the zeta function.

References

arXiv:hal-01984389v1: Probabilistic analysis of least common multiples.
Tenenbaum, G. (2015). Introduction to Analytic and Probabilistic Number Theory.
Selberg, A. (1946). Contributions to the theory of the Riemann zeta-function.