Fibonacci, Dirichlet, and Gauss in a single sum

Benoit Cloitre

Fibonacci, Dirichlet, and Gauss in a single sum

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that the remainder in the Fibonacci fractional-part sum is governed by the Gauss circle error for odd n and the Dirichlet divisor error for even n, making the optimal-exponent conjectures for the two classical problems equ

desk verdict A correct and useful bridge: the Fibonacci fractional-part sum carries the Gauss circle and Dirichlet divisor errors in its odd and even remainders, with an unconditional equivalence of error exponents. read the letter →

arxiv 2607.20960 v1 pith:LGZYH5WN submitted 2026-07-23 math.NT

Benoit Cloitre This is my paper

classification math.NT MSC 11B3911N3711P21

keywords FibonaccinumbersfractionalpartsGausscircleproblemDirichletdivisorerrorexponentsLucassequencesJacobitwo-squareidentitysums

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the sum of fractional parts of F_n/F_k for k=1..n. It proves that when n is odd, the remainder term is exactly a linear combination of the Gauss circle error term at n and n/2, up to O(n^ε). When n is even, the remainder is a linear combination of the Dirichlet divisor error term at n/2, n/4, and n/8. Consequently, the optimal exponent in the Fibonacci sum along odd indices is equivalent to the Gauss circle problem, and along even indices to the Dirichlet divisor problem. This unifies two classical lattice-point problems inside a single arithmetic sum.

What carries the argument

The argument rests on a periodic-profile reduction (Lemma 3.2): up to O(n^ε), F(n) equals a sum of the form Σ_{k≤n} w(⌊n/k⌋) or a combination of two such sums, where w, u, v are period-4 functions determined solely by the parity of k and ⌊n/k⌋ modulo 4. This reduction uses the exponential decay of Fibonacci ratios F_r/F_k and F_{k-r}/F_k. The periodic sums are then evaluated by Lemma 2.1, a telescoping identity that converts Σ ω(⌊x/k⌋) into a divisor sum Σ_{j≤x} Σ_{q|j}(ω(q)−ω(q−1)). For odd n, the resulting weight is the nonprincipal character χ_4, and Jacobi's two-square identity turns the divisor sum into the Gauss circle count N_C(x). For even n, the weights give the floor sum of the div

What would settle it

Find a sequence of n for which Σ_{h≥1} φ^{-h}(τ(n+h)+τ(n−h)) is not O(n^ε) — for example, compute this sum exactly for n = 2^m and n = 3^m and show it grows like a positive power of n; this would invalidate Lemma 3.2 and break the equivalence.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every odd n≥3, F(n) = (π/8)n + (1/4)(Δ_C(n) − Δ_C(n/2)) + O(n^ε), and for every even n≥4, F(n) = (3 log 2 /4)n + 2Δ_H(n/2) − 5Δ_H(n/4) + 2Δ_H(n/8) + O(n^ε). Together with Proposition 6.1, this gives an unconditional equivalence: a bound O(n^{θ+ε}) for the odd-indexed Fibonacci remainder holds if and only if Δ_C(x) = O(x^{θ+ε}), and similarly for even indices and Δ_H. In particular, the longstanding conjectures that 1/4 is optimal for both the Gauss circle and Dirichlet divisor problems are equivalent to the conjecture that 1/4 is the optimal remainder exponent for the Fibonacci sum, separately along odd and even indices. The paper also proves analogous f

Load-bearing premise

The entire transfer of exponents rests on the assumption that the sum of exponentially small Fibonacci remainders, weighted by the divisor function, is still O(n^ε); if that error were as large as n^{1/4}, the claimed equivalence would break.

Editorial extensions

If this is right

The conjectures that 1/4 is optimal in the Gauss circle problem and the Dirichlet divisor problem are equivalent to the conjecture that 1/4 is optimal for the remainder of F(n) along odd and even indices, respectively.
Any new bound on Δ_C or Δ_H immediately transfers to the corresponding Fibonacci sums, and conversely.
For the Lucas sequence, the parity assignment reverses: odd n connect to the Dirichlet divisor error, even n to the Gauss circle error.
The equivalence (6) holds unconditionally, so the two classical unsolved problems are now tied to the asymptotics of a single fractional-part sum.
The method extends to a family of second-order recurrences with determinant −1, showing the phenomenon is not unique to Fibonacci numbers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the optimal exponents for Δ_H and Δ_C turn out to be different, the Fibonacci sum would exhibit different optimal remainder exponents for odd and even n, giving a parity-dependent dichotomy in one sum — a scenario the paper does not discuss but which (6) implies.
Numerical computation of F(n) for large n could provide independent estimates of Δ_H and Δ_C at sparse points, because the identities are exact up to O(n^ε); this could complement direct lattice-point counts and test the 1/4 conjecture empirically.
For higher-order recurrences like Tribonacci, the paper's observations suggest that residue laws with multiple nontrivial residues might lead to new, non-classical counting functions; exploring this could reveal whether the connection to lattice-point problems is special to recurrences whose companion matrix has determinant −1.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper does what it says, and the main equivalence is real. It shows that the parity-split Fibonacci fractional-part sum F(n) has a remainder built from the Gauss circle error at odd n and from the Dirichlet divisor error at even n, up to O(n^ε). The MathOverflow answer [4] only had the two leading terms with O(√n); Theorem 1.1's remainder identities and the exponent equivalence in (6) are genuinely new. The generalized Fibonacci/Lucas results in Theorem 7.1 are a sensible extension, and the parity swap for Lucas numbers is a nice touch.

I checked the core algebra. Lemma 3.2's periodic profiles are consistent with the Fibonacci congruences; the manipulations with w, u, v in Sections 4–5 are correct; the linear main terms come out right. Proposition 6.1's converse is the part that makes the claim interesting, and it holds: the odd case reduces to a binary recurrence for Δ_C, and the even case is handled by the B(m) trick. The equivalence is not a conjecture—it is an unconditional iff between error exponents.

The soft spots are modest. The O(n^ε) estimates in Lemma 3.2 are the main residual uncertainty: individual fractional parts are close to the periodic profile only exponentially in the smaller index, and the accumulated error is bounded by sums of τ(n±h) with exponential weights. That is standard and plausible, but it is not machine-checked. There is also a tiny edge issue: formula (5) for even n uses Δ_H(n/8), which is not defined for n<8 unless one extends N_H(x)=0 for x<1; this is a trivial fix. The paper is a preprint with no accompanying code, so confidence should be moderate, not high. The citation pattern looks honest: the MathOverflow answer is credited, and the self-citation to a monograph in preparation is not used to support the main result. The Tribonacci remark is explicitly labeled as numerical and unproved, which is the right way to handle it.

Who should read it: anyone working on the Dirichlet divisor or Gauss circle problems, or on fractional parts of linear recurrences. It deserves a serious referee; I would send it out.

Referee Report

0 major / 5 minor

Summary. The paper studies the fractional-part sum F(n)=∑_{k=1}^n {F_n/F_k}. Using Fibonacci congruences modulo F_k, it shows that, up to O(n^ε), F(n) is a periodic profile sum depending only on ⌊n/k⌋ and the parity of k. For odd n the profile is the indicator of k odd and ⌊n/k⌋ ≡ 1,2 (mod 4); via Jacobi's two-square identity this becomes (N_C(n)−N_C(n/2))/4+O(n^ε), giving Theorem 1.1(4) with the Gauss circle error. For even n the profile becomes a combination of hyperbolic lattice counts N_H(n/2), N_H(n/4), N_H(n/8), giving Theorem 1.1(5) with the Dirichlet divisor error. Proposition 6.1 proves the converse implications, so admissible remainder exponents for F(n) along odd indices coincide with θ_C and along even indices with θ_H. Section 7 extends the identities to the one-parameter families F^{(c)} and L^{(c)} and observes that the Lucas sequence swaps the parity patterns. The paper also gives an elementary warm-up example (Proposition 2.2) in which a simple exponential sum already produces Δ_H.

Significance. If correct, the paper gives a striking, explicit bridge between a natural Fibonacci fractional-part sum and two classical lattice-point problems. The proof is elementary and self-contained: the main identities are derived from standard congruences, Jacobi's two-square theorem, and the divisor summatory function, with no fitted parameters and no circularity. The possible weak point identified in the stress-test note — the O(n^ε) error in Lemma 3.2 — does not land: the accumulated error is bounded by ∑_h φ^{-h}(τ(n+h)+τ(n−h))+O(τ(n)), which is O(n^ε) by the standard divisor bound and the convergence of ∑ φ^{-h}. The equivalence of error exponents is unconditional and, to my knowledge, new. The extension to generalized Fibonacci and Lucas sequences shows that the phenomenon is structural rather than an accident of the initial values. I regard the equivalence (6) as the main contribution.

minor comments (5)

[Section 6, proof of Proposition 6.1, odd case] After 'These estimates yield', the constant −π/2 appearing in Δ_C(m+1/2)=Δ_C(m)−π/2 is silently absorbed into the O(j^{θ+ε}) term. This is correct because θ+ε>0, but the absorption should be stated to avoid apparent loss of the constant.
[Sections 1 and 5] The linear main terms are written as 'π/8 n' and '3 log 2 /4 n'; parenthesized forms (π/8)n and (3 log 2 /4)n would improve readability.
[Section 7, after (22)–(23)] The claim that the constants in the error estimates may be chosen independently of c deserves one sentence of justification. From the Binet formulas, the relevant ratios are bounded by C α^{-h} with a constant C depending only on φ, since α≥φ and |β|/α≤φ^{-2}.
[Section 3, proof of Lemma 3.2] The sentence 'The final term O(τ(n)) accounts for the pairs with r=0' could be expanded slightly to note that this correction is needed for both parities of the quotient, because the periodic profiles may count a fractional part as 1 even when the actual term vanishes.
[Introduction, display (6)] The displayed equivalence with braces could be misread as a simultaneous equivalence. A short sentence clarifying that the C-bound and H-bound equivalences are proved separately would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and reduces to independent classical identities.

full rationale

The paper's central formulas (4) and (5) are derived from Lemma 2.1 (a telescoping identity), Lemmas 3.1–3.2 (Fibonacci congruences with an explicit exponentially weighted error bound), Jacobi's two-square identity, and the standard definitions of the Dirichlet divisor and Gauss circle error terms. None of these inputs presupposes the claimed connection to F(n); the parity profiles w, u, v are defined by an independent table, and the error term is bounded by sum_h phi^{-h}(tau(n+h)+tau(n-h)), which is O(n^epsilon) by the standard divisor estimate. Proposition 6.1 proves the converse implications by using the already-proved Theorem 1.1 in both directions; this is a genuine equivalence transfer, not a circular reduction, because it does not assume the bound being proved. The only self-citation, [2], is motivational ('arose in a broader study') and supports no proof step. There are no fitted parameters, no renamed empirical results, and no load-bearing self-citation chain. The Tribonacci remarks explicitly disclaim an asymptotic formula, so they introduce no circular support. Overall, the claimed equivalence of error exponents follows from explicit identities and standard estimates, and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters and no invented entities. The paper fits no constants to data; the only numbers appearing are structural coefficients such as π/8 and (3 log 2)/4 that arise from the asymptotics of N_C and N_H. The derivation relies entirely on standard Fibonacci/divisor identities and classical lattice-point error-term definitions.

assumptions (4)

standard math Divisor bound: for every ε>0, τ(j)=O(j^ε)
Used throughout Lemma 3.2, Proposition 2.2 and Proposition 6.1 to absorb all divisor-counting corrections into O(n^ε). Standard elementary estimate, but not proved in the paper.
standard math Jacobi's two-square identity: r₂(j)=4∑_{d|j}χ₄(d)
Invoked in Section 4 to convert the character-based floor sum into N_C(x). This is an external classical theorem; if it failed, the odd-index formula would not follow.
standard math Asymptotic normalizations N_H(x)=x log x+(2γ−1)x+Δ_H(x) and N_C(x)=πx+Δ_C(x), with Δ_H(x)=o(x)
Equations (2)–(3). These definitions are the target objects; the Δ_H(x)=o(x) property is used in the telescoping argument of Proposition 6.1 and follows from Dirichlet's classical bound.
standard math Standard Fibonacci identities: addition formula, Cassini's identity, and divisibility F_a|F_b whenever a|b
Used in Lemma 3.1 and Section 7 to derive the congruence F_{qk+r} ≡ ±F_r or ±F_{k−r} mod F_k. These are classical and proved in standard references, but the paper does not re-prove all of them.

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Cite this review

Pith. "Pith review of Fibonacci, Dirichlet, and Gauss in a single sum." pith.science (2026). https://pith.science/paper/LGZYH5WN

@misc{pith2026260720960,
  author       = {Pith},
  title        = {Pith review of: Fibonacci, Dirichlet, and Gauss in a single sum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGZYH5WN}},
  note         = {Machine review of arXiv:2607.20960}
}

abstract

We study the fractional-part sums $\sum_{k=1}^{n}\{F_n/F_k\}$, where $F_n$ is the $n$th Fibonacci number. Their asymptotic behavior depends on the parity of $n$. For odd $n$, the remainder is expressed in terms of the Gauss circle error term. For even $n$, it is expressed in terms of the Dirichlet divisor error term. Thus determining the optimal remainder exponent for the odd Fibonacci sums is equivalent to the Gauss circle problem, while the corresponding question for the even sums is equivalent to the Dirichlet divisor problem. We also prove analogous formulas for a family of second-order recurrences, including the Lucas sequence, for which the roles of the two parities are exchanged.

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Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [4]

Limit involving the fractional part and the Fibonacci numbers,

GH from MO, Answer to the question “Limit involving the fractional part and the Fibonacci numbers,”MathOverflow, March 30, 2024.https://mathoverflow.net/a/468048 11

2024

[2] [1]

B. C. Berndt, S. Kim, and A. Zaharescu, The circle problem of Gauss and the divisor problem of Dirichlet—still unsolved,Amer. Math. Monthly125(2018), 99–114. https: //doi.org/10.1080/00029890.2018.1401853

arXiv 2018

[3] [2]

Cloitre,Theory of regular arithmetic functions, vol

B. Cloitre,Theory of regular arithmetic functions, vol. I, Part IV, monograph in preparation

[4] [3]

P. G. L. Dirichlet, Über die Bestimmung der mittleren Werthe in der Zahlentheorie,Abh. Königl. Preuss. Akad. Wiss.(1849), 69–83

[5] [5]

R. K. Guy,Unsolved Problems in Number Theory, 3rd ed., Problem Books in Mathematics, vol. 1, Springer-Verlag, New York, 2004, Section F1, 365–367.https://doi.org/10.1007/ 978-0-387-26677-0

2004

[6] [6]

G. H. Hardy, On the expression of a number as the sum of two squares,Quart. J. Pure Appl. Math.46(1915), 263–283

1915

[7] [7]

G. H. Hardy, On Dirichlet’s divisor problem,Proc. London Math. Soc.(2)15(1916), 1–25

1916

[8] [8]

M. D. Hirschhorn, Partial fractions and four classical theorems of number theory, Amer. Math. Monthly107(2000), 260–264. https://doi.org/10.1080/00029890.2000. 12005191

arXiv 2000