Fibonacci, Dirichlet, and Gauss in a single sum

Technical Summary: Fibonacci, Dirichlet, and Gauss in a Single Sum

Problem Statement
This paper investigates the asymptotic behavior of the fractional-part sums defined by the Fibonacci sequence:
$$F(n) = \sum_{k=1}^{n} \left\{ \frac{F_n}{F_k} \right\}$$
where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part. The central problem is to determine the precise asymptotic expansion of $F(n)$ and to characterize the error terms associated with this sum. The study reveals that the behavior of $F(n)$ is fundamentally dependent on the parity of $n$ (odd vs. even), linking the sum to two distinct classical lattice-point problems: the Gauss circle problem and the Dirichlet divisor problem.

Methodology
The proof strategy combines number-theoretic congruences of Fibonacci numbers with techniques from analytic number theory, specifically exponential weighted divisor sums.

1. Quotient Identity: The authors establish a general identity (Lemma 2.1) relating sums of the form $\sum \omega(\lfloor x/k \rfloor)$ to divisor sums. This allows the transformation of fractional-part sums into sums involving arithmetic functions.
2. Fibonacci Congruences: A critical component is the analysis of $F_n \pmod{F_k}$. Using the division algorithm $n = qk + r$, the authors derive congruences for $F_{qk+r}$ modulo $F_k$. These congruences depend on the parity of the quotient $q$ and the remainder $r$, effectively determining whether the fractional part $\{F_n/F_k\}$ is close to 0 or 1.
3. Periodic Profiles: Based on the congruences, the fractional parts are shown to follow periodic patterns determined by $q \pmod 4$. The authors define periodic functions $w, u, v$ that capture these patterns.
4. Arithmetic Identities:
   • For odd $n$, the analysis reduces to a sum involving the non-principal character modulo 4, $\chi_4$. This connects the sum to Jacobi's two-square identity, which counts integer lattice points inside a circle ($N_C(x)$).
   • For even $n$, the analysis reduces to an alternating divisor sum, connecting the result to the count of lattice points under a hyperbola ($N_H(x)$).

Key Contributions and Results

• Theorem 1.1 (Main Result): The paper provides explicit asymptotic formulas for $F(n)$ that express the remainder terms directly in terms of the error functions of the Gauss circle and Dirichlet divisor problems.

  • For odd $n$:
    $$F(n) = \frac{\pi}{8}n + \frac{1}{4}\left( \Delta_C(n) - \Delta_C\left(\frac{n}{2}\right) \right) + O(n^\varepsilon)$$
    Here, $\Delta_C(x) = N_C(x) - \pi x$ is the error term for the Gauss circle problem.
  • For even $n$:
    $$F(n) = \frac{3 \log 2}{4}n + 2\Delta_H\left(\frac{n}{2}\right) - 5\Delta_H\left(\frac{n}{4}\right) + 2\Delta_H\left(\frac{n}{8}\right) + O(n^\varepsilon)$$
    Here, $\Delta_H(x) = N_H(x) - x \log x - (2\gamma - 1)x$ is the error term for the Dirichlet divisor problem.

• Equivalence of Error Exponents: The paper establishes an unconditional equivalence between the optimal error exponents of the classical problems and the Fibonacci sum. If $\Delta_C(x) = O(x^{\theta_C + \varepsilon})$ and $\Delta_H(x) = O(x^{\theta_H + \varepsilon})$, then the remainder of $F(n)$ is $O(n^{\theta + \varepsilon})$ where $\theta = \theta_C$ for odd $n$ and $\theta = \theta_H$ for even $n$. Consequently, determining the optimal exponent for the Fibonacci sum is equivalent to solving the respective classical problems.

• Generalization to Lucas and Second-Order Recurrences:
  The results are extended to a family of second-order recurrences $F^{(c)}_j$ and associated Lucas sequences $L^{(c)}_j$.

  • For the generalized Fibonacci sequence, the parity dependence remains the same as the standard Fibonacci case.
  • For the generalized Lucas sequence, the roles of the parities are exchanged: odd indices select the Dirichlet divisor error term, while even indices select the Gauss circle error term.

• Tribonacci Observation: The paper briefly examines the Tribonacci sequence. Numerical computations suggest a different behavior (leading coefficient near $1/2$) and note that the matrix reduction modulo $T_k$ involves multiple residues rather than a scalar sign, implying that a simple analogue to the Fibonacci/Lucas results may not exist without accounting for more complex residue data.

Significance
The paper's primary significance lies in unifying two historically parallel and distinct problems in analytic number theory—the Gauss circle problem and the Dirichlet divisor problem—within a single, elementary-looking sum involving Fibonacci numbers.

The authors demonstrate that the parity of the index $n$ acts as a switch, selecting which classical error term governs the asymptotic behavior of the sum. This provides a new perspective on the "optimal remainder exponent" question: proving a bound for the Fibonacci sum along odd (or even) indices is mathematically equivalent to proving the same bound for the Gauss circle (or Dirichlet divisor) problem. The work refines previous partial results (such as those found on MathOverflow) by expressing the remainders exactly in terms of the standard error functions $\Delta_C$ and $\Delta_H$, rather than just providing a generic $O(\sqrt{n})$ bound.