The moments of split greatest common divisors

Technical Summary: The Moments of Split Greatest Common Divisors

Problem Statement
This paper addresses the asymptotic behavior of the moments of the sequence $(\gcd(n, u_n))_{n \in \mathbb{N}}$, where $(u_n)$ is a non-degenerate Lucas sequence. While the study of $\gcd(u_n, v_n)$ for general divisibility sequences is a difficult problem connected to deep conjectures in Diophantine geometry (such as Vojta's conjecture), the specific case of $\gcd(n, u_n)$ corresponds to divisibility sequences arising from the algebraic group $\mathbb{G}_a \times \mathbb{G}_m$.

Previous work by Sanna established the asymptotics for the logarithmic moments $\sum_{n \le x} (\log \gcd(n, u_n))^\lambda$. However, the moments of the sequence itself, $\sum_{n \le x} \gcd(n, u_n)^\lambda$, remained largely unresolved, with only partial upper bounds available (e.g., by Mastrostefano). The primary objective is to determine precise asymptotic estimates for these moments, providing both unconditional results and conditional results based on standard conjectures in analytic number theory.

Methodology
The authors employ a combination of analytic number theory, properties of Lucas sequences, and combinatorial constructions involving smooth numbers.

1. Rank of Appearance and Divisibility: The analysis relies on the rank of appearance $z(m)$, the smallest $n$ such that $m \mid u_n$. The condition $n' = \gcd(n'k, u_{n'k})$ is translated into conditions involving $z(n')$ and the divisibility of $n'k$. The sum is decomposed based on pairs $(n', k)$ where $n = n'k$.
2. Upper Bounds: The unconditional upper bound is derived by splitting the summation domain into cases based on the size of auxiliary variables. A critical component is bounding the number of integers $n$ for which $\gcd(n, z(n))$ is large. This utilizes a result by Gordon and Pomerance regarding the count of integers with a fixed rank of appearance.
   • For the conditional upper bound, the authors assume Conjecture 3.2, which posits a stronger density estimate for integers with a fixed rank of appearance, analogous to bounds for pseudoprimes.
3. Lower Bounds: The strategy involves constructing a large subset of integers $n$ such that $n \mid u_n$ (denoted as $N_u(x)$).
   • Conditional Lower Bound: The authors construct a set of integers composed of shifted primes $p$ where $p - (\Delta/p)$ is smooth. This relies on Conjecture 4.2, a weak instance of Granville's conjecture regarding the distribution of smooth shifted primes.
   • Unconditional Lower Bound: For the specific case where the coefficient $a_2 = \pm 1$, the authors utilize a recent result by Lichtman on the distribution of smooth shifted primes in arithmetic progressions (Theorem 5.1). This allows for the construction of a sufficiently large set of integers $n$ dividing $u_n$ without assuming unproven conjectures.
4. Integral Estimates: To transition from the cardinality of the set $N_u(x)$ to the moment sum, the authors apply Karamata's integral theorem and partial summation arguments.

Key Results

• Conditional Asymptotics (Theorem 1.3): Assuming Conjectures 3.2 and 4.2, for a non-degenerate Lucas sequence with $a_2 = \pm 1$, the moments satisfy:
  $$\sum_{n \le x} \gcd(n, u_n)^\lambda = x^{\lambda+1 - (1+o_\lambda(1)) \frac{\log \log \log x}{\log \log x}}$$
  This solves the moment problem for $\mathbb{G}_a \times \mathbb{G}_m$ under standard conjectures.

• Unconditional Upper Bound (Theorem 1.4): For any non-degenerate Lucas sequence, the authors prove:
  $$\sum_{n \le x} \gcd(n, u_n)^\lambda \le x^{\lambda+1 - (1/2+o_\lambda(1)) \frac{\log \log \log x}{\log \log x}}$$
  The exponent $1/2$ is noted as an artifact of the current method.

• Unconditional Lower Bound (Theorem 1.4): If $a_2 = \pm 1$, the authors establish a lower bound:
  $$\sum_{n \le x} \gcd(n, u_n)^\lambda \ge x^{\lambda + 0.715}$$
  This result is derived using Lichtman's theorem on smooth shifted primes.

• Refined Conjecture (Conjecture 1.5): Based on the analogy with Lucas pseudoprimes, the authors conjecture a more precise asymptotic form involving an exponential term with iterated logarithms, suggesting the current bounds are not the final limit of precision.

• Corollaries: The methods yield improved bounds on the distribution of $\gcd(n, u_n)$ (Corollary 1.6) and provide a new analytic proof for the main theorem in [25] regarding the count of $n$ such that $\gcd(n, u_n) = n$.

Significance and Claims
The paper claims to make "crucial progress" on the moment problem for $\mathbb{G}_a \times \mathbb{G}_m$. Specifically, it:

1. Solves Conjectures 3 and 5 of [29] in the case $a_2 = \pm 1$.
2. Solves the main conjecture of [13].
3. Improves the unconditional bound of [2, Th. 1.3] via Lichtman's bound.
4. Provides a matching lower bound for the conjectures in [13] and [25].
5. Answers a specific case of Question 1.1 and makes progress on Question 1.2 in [27].

The authors note that proving the conditional asymptotic (Theorem 1.3) unconditionally is currently infeasible, as the lower bound relies on the distribution of smooth shifted primes, a problem not yet resolved without assuming conjectures like Elliott–Halberstam. The case $a_2 = \pm 1$ is highlighted as the most "interesting" and tractable scenario for these methods.