Fibonacci Rank Weighted Prime Tail

Abstract

The weighted prime Fibonacci first-rank tail is summable with an explicit cutoff bound.

For each prime p, z(p) is the least positive d with p dividing F(d). The cutoff y is real. Define w(y,p) as 1/(p*z(p)) when p is prime and p exceeds y, and zero otherwise.

Theorem 1.1 (A uniform real-cutoff tail bound).

$$\begin{array}{r}\forall y\in \mathbb{R},y\ge 2\Rightarrow \\ \mathrm{Summable}\left(w(y,\cdot )\right)\wedge \sum _{p\in \mathbb{N}}w\left(y,p\right)\le \frac{16}{\sqrt{y}}\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/FibonacciRankWeightedPrimeTail.result (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Juan Jose Alba Gonzalez, Florian Luca, Carl Pomerance, and Igor E. Shparlinski (2012). On numbers n dividing the nth term of a linear recurrence. DOI: 10.1017/S0013091510001355. URL: https://doi.org/10.1017/S0013091510001355.

Commentary.

The sum over all natural p converges, and the displayed bound holds for every real y at least two. The first-zero buckets at ranks one through five are empty, empty, the singleton two, the singleton three, and the singleton five, respectively.

On a twofold interval with lower endpoint Y, split the primes according to whether z(p) is at most the square root of Y. For small ranks, a rank-d bucket has fewer than d primes, so each rank contributes at most 1/Y. There are at most the square root of Y such ranks. For large ranks, each term is at most 1/(Y*sqrt(Y)), and the interval has at most 2Y natural numbers. The total is at most 3/sqrt(Y).

Partitioning every finite prime sum into twofold intervals bounds it by a geometric series with ratio 1/sqrt(2). Its ratio is at most three quarters, giving a uniform bound of 12/sqrt(y), hence the displayed 16/sqrt(y). Nonnegative finite sums with this common bound establish summability and the infinite-sum inequality together.

The decay order y to the power minus one half is known from the twofold-interval argument in the proof of Theorem 1.2 in Alba Gonzalez, Luca, Pomerance and Shparlinski, On numbers n dividing the nth term of a linear recurrence, Proceedings of the Edinburgh Mathematical Society 55 (2012), 271-289. The explicit constant for all real y at least two is the FIB volume’s deduction. No novelty claim is made.

References

• Truth anchor: D5/S3/Arith/Robin/FibonacciRankWeightedPrimeTail.result
• Dependency: D5/S3/Arith/FibonacciAtomic/TimeSampling
• Dependency: D5/S3/Arith/Robin/FibonacciRankEulerTail