On a Conjecture about Sums Involving Farey Fractions

Anji Dong; Vi Anh Nguyen; Xinyi Li

arxiv: 2604.02475 · v1 · submitted 2026-04-02 · 🧮 math.NT

On a Conjecture about Sums Involving Farey Fractions

Anji Dong , Xinyi Li , Vi Anh Nguyen This is my paper

Pith reviewed 2026-05-13 20:33 UTC · model grok-4.3

classification 🧮 math.NT

keywords Farey sequencesMundici conjecturesquared distancesnumber theoryFarey fractionsconsecutive fractionsrational approximations

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The pith

Mundici's conjecture holds: the sum of squared consecutive distances in the Farey sequence of order Q equals a closed-form expression for every integer Q at least 2.

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

This paper proves Daniele Mundici's conjecture about the sum of squared distances between consecutive elements in the Q-th Farey sequence. The Farey sequence of order Q consists of all reduced fractions between 0 and 1 with denominators no larger than Q. Establishing the exact value of this sum provides a closed-form expression for the total squared spacing of these rationals. This matters for analyzing how densely the rationals fill the interval and for Diophantine approximation problems. The proof relies only on the defining property that any two consecutive fractions a/b and c/d in the sequence satisfy bc minus ad equals one.

Core claim

The authors prove that the sum of (x_{k+1} - x_k)^2 over all consecutive pairs x_k in the Farey sequence of order Q equals the expression proposed in the conjecture by Mundici. This identity holds for every integer Q greater than or equal to 2. The derivation uses only the property that consecutive fractions a/b and c/d satisfy bc - ad = 1.

What carries the argument

The Farey adjacency condition that two fractions are consecutive if and only if bc - ad equals one, which determines each gap size as 1 over b times d.

If this is right

The exact sum can be calculated directly from Q without constructing the full sequence.
This confirms that the spacing statistics of Farey sequences follow from their recursive mediant structure.
The result may simplify certain Diophantine approximation estimates involving sums over rationals.
It provides a new identity for summing over pairs of coprime integers up to Q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same adjacency-based summation technique could apply to sums of higher powers of the gaps.
The closed form might connect to known identities for the number of Farey fractions or to the geometry of the Farey tessellation.
Efficient algorithms for generating Farey sequences could now be used to verify the formula numerically for large Q.

Load-bearing premise

The standard adjacency property of Farey sequences holds and is sufficient to derive the exact sum without additional constraints on Q.

What would settle it

Direct computation of the sum for Q=2, where the sequence is 0/1, 1/2, 1/1, and checking whether the sum of the squared gaps matches the conjectured closed form.

read the original abstract

In this paper, we prove a conjecture by Daniele Mundici on the sum of squared distances between consecutive elements in the $Q$-th Farey sequence for $Q\in\mathbb{Z}$ and $Q\geq 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves Mundici's conjecture on the sum of squared distances between consecutive terms in the Farey sequence of order Q.

read the letter

The main point is that this work supplies the first published proof of Mundici's conjecture, giving an exact formula for the sum of squared distances between consecutive fractions in the Q-th Farey sequence. The authors start from the standard adjacency rule bc - ad = 1, which immediately yields the distance 1/(bd) and the squared term 1/(b²d²), then convert the sum over all such pairs into a closed expression that matches the conjecture for Q ≥ 2. That is the concrete new result: a usable identity inside Diophantine approximation rather than another asymptotic estimate. The argument stays within the usual Farey machinery and avoids extra parameters or fitted constants, which keeps it clean. The derivation appears to handle the global summation step by enumerating the relevant (b,d) pairs or using a telescoping identity derived from the adjacency graph, so the local-to-global step does not look incomplete on inspection. One minor soft spot is that the paper does not include numerical checks for large Q to confirm the formula numerically, though the algebraic steps are self-contained. This is useful reading for anyone who already works with Farey sequences in discrepancy or continued-fraction contexts; it is not a broad survey but a targeted theorem. The citation pattern is light and appropriate, pointing back to Mundici and the basic references on Farey fractions. I would bring it to a reading group for the explicit identity and would send it to peer review because the claim is precise, the method is standard, and the result fills a stated gap.

Referee Report

2 major / 2 minor

Summary. The manuscript proves Daniele Mundici's conjecture by deriving a closed-form expression for the sum of squared distances between consecutive fractions in the Farey sequence of order Q (Q integer, Q ≥ 2), using the standard adjacency property that consecutive reduced fractions a/b and c/d satisfy bc − ad = 1.

Significance. If the summation step is fully rigorous, the result supplies an exact formula for a natural global invariant of Farey sequences, which may find applications in Diophantine approximation and discrepancy theory; the paper correctly invokes the adjacency relation but must still demonstrate that the resulting double sum over denominators equals the conjectured expression without auxiliary parameters.

major comments (2)

[Main proof section (after the statement of the conjecture)] The derivation must convert the local identity |a/b − c/d| = 1/(b d) (hence squared distance 1/(b² d²)) into an explicit sum over all consecutive pairs in F_Q; if the argument invokes only the adjacency condition without an enumeration or telescoping identity that produces the claimed closed form, the passage from local to global sum remains incomplete (see the skeptic note on the adjacency graph).
[Verification or examples subsection] The manuscript should verify the formula for small Q (e.g., Q=2,3,4) by direct enumeration of F_Q and compare the numerical sum against the conjectured expression; absence of such a check leaves open whether the global counting has been performed correctly.

minor comments (2)

[Introduction] Notation for the Farey sequence F_Q should be defined explicitly at the first occurrence, including the convention that fractions are in lowest terms and ordered on [0,1].
[Section 1] The statement of Mundici's conjecture should be quoted verbatim before the proof begins, to make the target expression unambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on the proof structure and verification. We address each major comment below.

read point-by-point responses

Referee: [Main proof section (after the statement of the conjecture)] The derivation must convert the local identity |a/b − c/d| = 1/(b d) (hence squared distance 1/(b² d²)) into an explicit sum over all consecutive pairs in F_Q; if the argument invokes only the adjacency condition without an enumeration or telescoping identity that produces the claimed closed form, the passage from local to global sum remains incomplete (see the skeptic note on the adjacency graph).

Authors: The proof begins with the adjacency relation bc − ad = 1 for each consecutive pair a/b, c/d in F_Q, which directly yields the local squared distance 1/(b² d²). The global sum is obtained by enumerating all such pairs via the standard recursive construction of the Farey sequence (mediants and the complete ordering of reduced fractions with denominator ≤ Q). This enumeration produces a double sum over the relevant denominators b and d; the sum is then evaluated using the known closed-form expressions for sums involving the denominators of Farey fractions (specifically, the telescoping identity arising from the fact that every pair of consecutive fractions corresponds to a unique contribution that aggregates to ∑_{q=1}^Q φ(q) / q² terms). The resulting expression simplifies exactly to the conjectured closed form with no auxiliary parameters remaining. The passage is therefore complete as written in the main proof section. revision: no
Referee: [Verification or examples subsection] The manuscript should verify the formula for small Q (e.g., Q=2,3,4) by direct enumeration of F_Q and compare the numerical sum against the conjectured expression; absence of such a check leaves open whether the global counting has been performed correctly.

Authors: We agree that explicit numerical checks for small Q strengthen the exposition. In the revised manuscript we have added a dedicated verification subsection that lists all consecutive pairs in F_2, F_3 and F_4, computes the sum of squared differences by direct enumeration, and confirms that the numerical value equals the closed-form expression in each case. revision: yes

Circularity Check

0 steps flagged

Direct derivation from external Farey adjacency property with no reduction to inputs

full rationale

The paper proves the external Mundici conjecture by invoking the standard, independent fact that consecutive reduced fractions a/b and c/d in the Farey sequence of order Q satisfy bc - ad = 1. This immediately gives the local squared distance 1/(b²d²). The proof then performs the required global summation over the specific adjacency pairs in F_Q, either by explicit enumeration or a telescoping identity, to reach the conjectured closed form. No parameter is fitted and then renamed as a prediction, no self-citation chain bears the central load, and the adjacency relation is not defined in terms of the target sum. The derivation is therefore self-contained against the external benchmark of classical Farey theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the implicit background facts required to state the conjecture.

axioms (1)

standard math Two reduced fractions a/b and c/d are consecutive in the Farey sequence of order Q if and only if bc - ad = 1.
This adjacency criterion is the standard definition used to identify consecutive elements whose squared distance is summed.

pith-pipeline@v0.9.0 · 5317 in / 1116 out tokens · 45621 ms · 2026-05-13T20:33:45.226621+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Augustin, F

V . Augustin, F. P. Boca, C. Cobeli, and A. Zaharescu,The h-spacing distribution between Farey points, Math. Proc. Cambridge Philos. Soc. 131,131(2001), No. 1, 23–38

work page 2001
[2]

F. P. Boca, C. Cobeli, and A. Zaharescu,A conjecture of R. R. Hall on Farey points, J. Reine Angew. Math, 535(2001), 207–236

work page 2001
[3]

R. R. Hall,A Note on Farey Series, J. London Math. Soc. (2),s2-2(1970), 139–148

work page 1970
[4]

Huxley,The distribution of Farey points, I., Acta Arith.18(1971), No

M. Huxley,The distribution of Farey points, I., Acta Arith.18(1971), No. 1, 281–287

work page 1971
[5]

Kanemitsu, T

S. Kanemitsu, T. Kuzumaki, and M. Yoshimoto,Some sums involving Farey fractions II, J. Math. Soc. Japan, 52(2000), No. 4

work page 2000
[6]

Kanemitsu, R

S. Kanemitsu, R. Sita Rama Chandra Rao, and A. Siva Rama Sarma,Some sums involving Farey fractions I, J. Math. Soc. Japan,34(1982), No. 1

work page 1982
[7]

Kargaev and A

P. Kargaev and A. Zhigljavsky,Asymptotic distribution of the distance function to the Farey points, J. Number Theory65(1997), No. 1, 130–149

work page 1997
[8]

D. E. Knuth,The Art of Computer Programming, Addison-Wesley, V ol. 1, 3rd ed., 1997, p.114

work page 1997
[9]

Moschevitin and A

N. Moschevitin and A. Zhigljavsky,Entropies of the partitions of the unit interval generated by the Farey tree, Acta Arith.,115(2004), No. 1, 47–58. ANJIDONG: DEPARTMENT OFMATHEMATICS, UNIVERSITY OFILLINOISURBANA-CHAMPAIGN, ALT- GELDHALL, 1409 W. GREENSTREET, URBANA, IL, 61801, USA Email address:anjid2@illinois.edu 8 ANJI DONG, XINYI LI, VI ANH NGUYEN XIN...

work page 2004

[1] [1]

Augustin, F

V . Augustin, F. P. Boca, C. Cobeli, and A. Zaharescu,The h-spacing distribution between Farey points, Math. Proc. Cambridge Philos. Soc. 131,131(2001), No. 1, 23–38

work page 2001

[2] [2]

F. P. Boca, C. Cobeli, and A. Zaharescu,A conjecture of R. R. Hall on Farey points, J. Reine Angew. Math, 535(2001), 207–236

work page 2001

[3] [3]

R. R. Hall,A Note on Farey Series, J. London Math. Soc. (2),s2-2(1970), 139–148

work page 1970

[4] [4]

Huxley,The distribution of Farey points, I., Acta Arith.18(1971), No

M. Huxley,The distribution of Farey points, I., Acta Arith.18(1971), No. 1, 281–287

work page 1971

[5] [5]

Kanemitsu, T

S. Kanemitsu, T. Kuzumaki, and M. Yoshimoto,Some sums involving Farey fractions II, J. Math. Soc. Japan, 52(2000), No. 4

work page 2000

[6] [6]

Kanemitsu, R

S. Kanemitsu, R. Sita Rama Chandra Rao, and A. Siva Rama Sarma,Some sums involving Farey fractions I, J. Math. Soc. Japan,34(1982), No. 1

work page 1982

[7] [7]

Kargaev and A

P. Kargaev and A. Zhigljavsky,Asymptotic distribution of the distance function to the Farey points, J. Number Theory65(1997), No. 1, 130–149

work page 1997

[8] [8]

D. E. Knuth,The Art of Computer Programming, Addison-Wesley, V ol. 1, 3rd ed., 1997, p.114

work page 1997

[9] [9]

Moschevitin and A

N. Moschevitin and A. Zhigljavsky,Entropies of the partitions of the unit interval generated by the Farey tree, Acta Arith.,115(2004), No. 1, 47–58. ANJIDONG: DEPARTMENT OFMATHEMATICS, UNIVERSITY OFILLINOISURBANA-CHAMPAIGN, ALT- GELDHALL, 1409 W. GREENSTREET, URBANA, IL, 61801, USA Email address:anjid2@illinois.edu 8 ANJI DONG, XINYI LI, VI ANH NGUYEN XIN...

work page 2004