arXiv:2507.00228v3

Distribution of Farey fractions with $k$ -free denominators

Bittu Chahal, Tapas Chatterjee, Sneha Chaubey

math.NT11B5711J7111K3811L0711L1511M26

Paper metadata

Original submission: June 30, 2025
Latest revision: September 2, 2026 · version 3
Claimed MSC: 11B57, 11J71, 11K38, 11L07, 11L15, 11M26
arXiv categories: math.NT

Abstract

We investigate the distributional properties of the sequence of Farey fractions with $k$ -free denominators in residue classes, defined as $\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.$ We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ , particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$ -functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ . Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ( $ν\ge 2$ ) of correlation measure. Specifically, we show the existence of the limiting pair ( $ν=2$ ) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains.

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Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated October 24, 2026

01Statements6 reported findingsSuspected incorrect statements

The uniform-frequency bound, the second-moment identity outside the residue class one, and the displayed pair-correlation formula are false. Equidistribution, discrepancy order, the joint-family GRH criterion, moment bounds and correlation existence survive the corrections described below.

Theorem 1.1Suspected incorrect statement

The asserted uniform Weyl bound fails at a common denominator

PDF p. 3 · Theorem 1.1; proof (25), p. 24 · arXiv:2507.00228v3

The theorem states the bound with a constant depending on $m$ , not on the nonzero frequency $r$ . Fix $m=b=1$ and any $k\ge2$ , and take $r=Q!$ . Every denominator in the Farey set divides $r$ , so every summand $e(r\gamma)$ equals one. The sum is therefore $N(Q,k,1)\sim\mathcal C(k,1)Q^2$ , with $\mathcal C(k,1)>0$ . The claimed right side is $O(Q^2\exp(-c(\log Q)^{3/5}/(\log\log Q)^{1/5}))=o(Q^2)$ , a contradiction. The fixed-frequency bound does not have this counterexample, but is weaker than the stated uniform minimum. The proof loses the dependence of the cancellation factor on $Q/q$ before (25). A verified safe uniform form replaces the minimum by $\min\{Q^2,|r|^\varepsilon Q\exp(-c'(\log Q)^{3/5}/(\log\log Q)^{1/5})\}$ , with a suitably smaller positive $c'$ . For the second bound split frequencies according as $\log|r|$ exceeds a suitable multiple of the decay exponent: in the large case a divisor bound absorbs that decay into $|r|^\varepsilon$ , and in the small case every divisor $q\mid r$ is below $Q^{1/2}$ , where the weighted Mobius estimate retains uniform decay. This changes the stated uniform bound; it is not a proof of the original formula.

Theorem 1.4 exact identitySuspected incorrect statement

The constant term is wrong when the Farey set excludes one

PDF p. 4 · first displayed identity in Theorem 1.4; Lemma 2.6, p. 15; (30), p. 26 · arXiv:2507.00228v3

Take $m=3$ , $b=2$ , $Q=2$ and any $k\ge2$ . The set is $\{1/2\}$ , its cardinality is one, and the left side is $1/4$ . On the right, $M_1(2)=-1$ and $M_2(1)=1$ , so the double sum is $1+1-2(1/2)=1$ . The printed right side is zero. This is not only a small-order exception. Put $\eta=1$ when $b\equiv1\pmod m$ and zero otherwise. Mobius inversion gives $\sum_qM_q(Q/q)=\eta$ , and the sawtooth sum between successive fractions is $Nu-v-\eta/2$ . If $\eta=0$ , reflection symmetry gives $\sum_j(\gamma_j-j/N)=-1/2$ and direct integration gives $I=N\sum_j(\gamma_j-j/N)^2-1/6$ . If $\eta=1$ , the printed relation $I=N\sum_j(\gamma_j-j/N)^2+1/12$ is valid. Thus a verified repair replaces the final $-1$ in the theorem's numerator by $2-3\eta$ . For every non-one residue class the error in the stated formula is exactly $1/(4N)$ . The asymptotic bounds are not invalidated by this correction.

Theorem 1.7Suspected incorrect statement

The displayed pair density omits the modulus factor

PDF p. 8 · (9) and the explicit formula for $F_k(n)$ ; Lemma 2.8, p. 16; (59)–(60), pp. 42–43 · arXiv:2507.00228v3

The local factor in Lemma 2.8 for $r_1=r_2=m$ includes $D_m=\frac{\varphi(m)^2}{m^2}\prod_{p\mid m}(1-p^{-2})^{-1}.$ Formula (60) still defines $F_k$ using that full factor, but the explicit $F_k$ in the theorem omits it while retaining $\varphi(m)^{-2}$ outside. A decisive regime is $m=2$ and $\mathcal C(k,2)<\lambda<2\mathcal C(k,2)$ , where only $n=\delta=d_1=d_2=r=1$ contributes. Write $A_k=\prod_{p\ne2}(1-2/(p^{k-1}(p+1)))>0$ . The density of coprime, odd, $k$ -free denominator pairs is $\frac14\prod_{p\ne2}(1-p^{-2})\left(1-\frac{2}{p^{k-1}(p+1)}\right)=\frac{2}{\pi^2}A_k.$ The theorem instead uses $6A_k/\pi^2$ , making its first nonzero density three times the actual density. The same change of variables as in (59) yields the positive multiplier $\lambda^{-2}\log(\lambda/\mathcal C(k,2))$ , so the discrepancy survives in the limiting correlation measure, not only in an intermediate count. Independent exact-fraction enumerations for $k=m=2$ on raw differences $(1.2/Q^2,1.8/Q^2)$ give normalized counts $0.088091$ , $0.087518$ and $0.085402$ at $Q=300,600,1200$ ; the correctly normalized limit is approximately $0.086346$ , whereas the printed formula gives approximately $0.259038$ . The local-density argument, rather than these finite samples, proves the contradiction.

Theorem 1.7, including the unrestricted modulus caseSuspected incorrect statement

Non-k-free coefficient tuples are also included incorrectly

PDF p. 42 · application of Lemma 2.8 with $\delta_1=d_1\delta$ , $\delta_2=d_2\delta$ ; (60) · arXiv:2507.00228v3

Lemma 2.8 assumes both $\delta_1$ and $\delta_2$ are $k$ -free. The proof applies it to $d_1\delta,d_2\delta$ without those restrictions. For $k=2$ , $\delta=d_1=d_2=2$ and $r=1$ , the original summand in (57) is identically zero since $4\mid q_i d_i\delta$ . Nevertheless the extended Euler product in the theorem has local factor at two equal to $1-(4+4)/(2(2+1))=-1/3$ . This affects $F_2(8)$ even for $m=1$ . For indices below eight there is no such nonzero erroneous tuple; all other non-square-free tuples at eight have a zero Euler factor already. Hence on $8\mathcal C(2,1)<\lambda<9\mathcal C(2,1)$ the missing contribution is exactly a positive multiple of $\lambda^{-2}\log(\lambda/(8\mathcal C(2,1)))$ , proving a discrepancy independently of the modulus-factor defect. A verified repair of the coefficient sum retains only $d_1\delta,d_2\delta$ that are $k$ -free and retains $D_m$ . The support condition bounds all four coefficient variables independently of $Q$ ; Lemma 2.8 is then applicable with constants allowed to depend on those finitely many fixed twists. This establishes existence and a corrected density, not the printed formula.

Corollary 1.2, Theorems 1.3 and 1.5, and moment boundsNo issue found

The global distribution conclusions have verified arguments

PDF pp. 3–5 and 24–30 · arXiv:2507.00228v3

For each fixed nonzero integer $r$ , the exact Ramanujan-sum expansion gives $|\sum_\gamma e(r\gamma)|\le Q\tau(|r|)$ . With $N\asymp_m Q^2$ , this already proves equidistribution without the false uniform minimum. The discrepancy argument uses $|M_q(Q/q)|\ll_m(Q/q)W(q)E(Q/q)$ , where $W(q)=\prod_{p\mid q}(1-p^{-1/2})^{-1}$ and a weakened cancellation factor $E(x)=\exp(-c\sqrt{\log x})$ suffices. The mean bound $\sum_{q\le x}W(q)\mu_k(q)^2\ll x$ and a dyadic split give $\sum_q|M_q(Q/q)|\ll_m Q$ , hence discrepancy $O_m(1/Q)$ ; the first or last fraction-free interval supplies the lower order. Under GRH the weighted Mobius bound gives $I\ll Q^{1+\varepsilon}$ , and the endpoint correction above is bounded, so Cauchy–Schwarz gives the claimed first-moment estimate. Read as a criterion for the family over reduced residue classes, the converse supplies every $M_b(x)$ bound; character orthogonality then gives $\sum_{n\le x}\mu(n)\chi(n)=O(x^{1/2+\varepsilon})$ , the reciprocal- $L$ holomorphy criterion for GRH. Taking all moduli gives the full Dirichlet GRH criterion. The unconditional second-moment bound also follows directly from the weighted Mobius estimate: group the double sum by $\gcd(q_1,q_2)=\delta$ , use $W(\delta a)\le W(\delta)W(a)$ and split $a$ and $\delta$ at square roots. The tails are power-saving and the remaining terms retain the stated subexponential decay. Thus the erroneous Proposition 2.3 below is unnecessary for that bound.

Theorem 1.6 and correlation existenceNo issue found

The lattice method supports the existence assertions

PDF pp. 6–7 and 20–23, 31–39 · arXiv:2507.00228v3

For $\nu<k$ , the Fourier expansion and Poisson summation reduce the statistic to finitely many coefficient tuples for each compact test-function support. Mobius inversion detects both coprimality and all $k$ -free linear forms. The lattice has rank two, and because its index involves only primes dividing the $d_j$ , it is coprime to $m$ in the contributing terms; consequently the reduction modulo $m$ is invertible and the congruence density is the stated $\#A_{m,r}/m^2$ . The determinant lower bound gives an absolutely convergent majorant with Euler factors $1+p^{-2}+(2^\nu-1)p^{-k}+(2^\nu-1)p^{-k-1}$ . For the convex polygonal regions used here, elementary weighted lattice counting, with an integral lattice basis chosen in reduced form, gives the summed error $O(Q^{1+\nu/k}\log Q)$ ; divided by $N\asymp Q^2$ it tends to zero exactly under $\nu<k$ . The final summation bounds require the index correction recorded below. The undefined extra $\Psi_k$ condition in the intermediate enlarged region on p. 38 must be omitted: all arithmetic $k$ -free restrictions are already imposed by the Mobius weights, and the ensuing region and final formula contain no such real-variable condition. With that local notation correction, the support argument justifies the region enlargement. The pair case for $k=2$ uses its separate proof after the two corrections above. Finally, the minimum nonzero normalized spacing tends to the positive constant $\mathcal C(k,m)$ , so none of these correlation measures is Poissonian.

02Proofs7 reported findingsSuspected incomplete or incorrect proofs

The false formulas arise from a lost frequency dependence, a missing endpoint term and two incorrect local-density manipulations. A separate harmonic-sum proposition also misses a principal-character normalization and nonprincipal residues.

Theorem 1.1 proofSuspected incorrect as written

The decay cannot be made uniform over all divisor sizes

PDF p. 24 · line preceding and display (25) · arXiv:2507.00228v3

Proposition 2.2 is applied at $x=Q/q$ . Its decay is controlled by $\log(Q/q)$ , which can remain bounded when $q$ is comparable to $Q$ . Replacing it by decay in $\log Q$ loses the required frequency dependence; summing over all divisors of a common multiple gives the counterexample in Part 1. Verified repairs for fixed frequency and for a differently capped uniform estimate are given there. Neither proves the printed uniform bound.

Lemma 2.6 and the moment identitySuspected incorrect as written

The sawtooth constant depends on whether one is included

PDF pp. 15 and 25–27 · Lemma 2.6 and (30) · arXiv:2507.00228v3

The final constant in the proof is $-\frac12\sum_qM_q(Q/q)$ , not always $-1/2$ . By Lemma 2.5 with the constant function at the endpoint, the sum of these coefficients is $\eta=\mathbf1_{b\equiv1\pmod m}$ . The verified sawtooth and integrated corrections are supplied in Part 1. They repair every first- and second-moment estimate because the change is $O(1)$ in $I$ and $O(1/N)$ in the moment; they do not leave the exact central identity unchanged.

Proposition 2.3Suspected incorrect as written

The harmonic progression main term has the wrong normalization and omits constants

PDF pp. 13–14 · formula for $\mathcal M_{m,d,l}$ and nonprincipal-character contour step · arXiv:2507.00228v3

Already for $d=l=1$ , the correct coefficient of $\log x$ in $\sum_{n\le x,\ n\equiv b\pmod m}\mu_k(n)^2/n$ is $1/(mL(k,\chi_0))$ . Character orthogonality contributes $1/\varphi(m)$ and the principal $L(s+1,\chi_0)$ residue is $\varphi(m)/m$ . The displayed formula instead has coefficient $\varphi(m)/(mL(k,\chi_0))$ ; for $m=3$ it is twice the correct leading term. Moreover, for each nonprincipal character the Perron integrand $F(s)x^s/s$ generally has residue $F(0)$ at zero even though $F(s)$ itself is regular. These constants depend on $b$ and are omitted. A correct asymptotic must divide the principal main term by $\varphi(m)$ and add the nonprincipal $\overline\chi(b)F_\chi(0)/\varphi(m)$ contributions. The second-moment upper bounds have the independent verified repair in Part 1, so no falsehood of those bounds is inferred from this proposition.

Theorem 1.7 proofSuspected incorrect as written

The two local-density errors propagate into the central formula

PDF pp. 41–43 · (57)–(62), especially (59)–(60) · arXiv:2507.00228v3

The modulus factor $D_m$ in the valid $P^k_{m,m}$ calculation is absent from the explicit theorem, and the application of Lemma 2.8 includes twists outside its $k$ -free hypotheses. The counterexamples and exact affected regimes are given in Part 1. Verified repair: restrict the coefficient sum to $\mu_k(d_1\delta)^2=\mu_k(d_2\delta)^2=1$ and use the complete $P^k_{m,m}$ factor. For every compact support, $\delta d_1d_2r<\Lambda/\mathcal C(k,m)$ bounds these twists independently of $Q$ , so their individual lattice errors and nonprincipal character-sum errors are $o(Q^2)$ after summation. The integral change of variables contributes the printed logarithmic kernel and no further normalization. This gives a valid limiting density with corrected coefficients, while (9) as printed remains false.

Theorem 1.6 summation boundsSuspected typo

The bounds assigned to the two coefficient indices are interchanged

PDF p. 7 · Theorem 1.6; pp. 34–35, 39 · arXiv:2507.00228v3

The theorem prints $A_j\le(\nu-1)C(\Lambda,k,m)^2$ and $B_j\le\nu C(\Lambda,k,m)^2$ . The proof defines $A_j=n_je_j$ and $B_j=n_j\Delta_j$ , with $e_j\le\nu C(\Lambda,k,m)$ , $\Delta_j\le(\nu-1)C(\Lambda,k,m)$ and $n_j\le C(\Lambda,k,m)$ . Thus the mechanically determined bounds are $A_j\le\nu C(\Lambda,k,m)^2$ and $B_j\le(\nu-1)C(\Lambda,k,m)^2$ . Interchanging these two index bounds makes the final expression match the established coefficient ranges and leaves the existence proof unchanged. In the intermediate region on p. 38, the extra undefined condition $\Psi_k(n_j(ye_j-x\Delta_j))=1$ is also extraneous; removing it gives precisely the enlarged region used in the next lines, with arithmetic restrictions already included in the coefficient weights.

Theorem 1.5 rangeSuspected minor formal issue

The discrepancy requires a nonempty set

PDF p. 5 · phrase for all orders in Theorem 1.5 · arXiv:2507.00228v3

When $m=3$ , $b=2$ and $Q=1$ , the Farey set is empty, so the displayed discrepancy with normalization by its cardinality is undefined. State the assertion for orders with $N(Q,k,m)>0$ , or for all sufficiently large $Q$ with fixed $m,b$ . This local range correction is consistent with every application, all of which concern the limit as $Q\to\infty$ ; the verified upper and lower discrepancy orders are unchanged.

Remaining global and lattice inputsNo issue found in the reviewed proof

The valid ingredients and limiting ranges are checked separately

PDF pp. 9–12, 14–24 and 31–39 · arXiv:2507.00228v3

The cardinality Euler product has its principal pole at two and gives the constant (8). The weighted Mobius series is a reciprocal Dirichlet $L$ -function times finite Euler factors, so the fixed-modulus zero-free region in Khale, Theorem 1.1 supplies the unconditional cancellation; the fixed-modulus small-height range is absorbed by choosing a smaller constant. Under GRH, holomorphy and standard reciprocal- $L$ growth in $\Re s>1/2$ supply the stated weakened bound. Here analytic continuation, not the incorrect phrase absolute convergence in that half-plane, is what the contour argument uses. The exact Farey expansion in Lemma 2.5 and its Ramanujan-sum specialization are valid. The sawtooth integral is $\gcd(q_1,q_2)^2/(12q_1q_2)$ ; grouping by the gcd gives convergent sums under the GRH exponent. In the correlation proof, the Poisson divisibility equations and support bounds agree with the underlying method of Boca–Zaharescu, §§3–4. Absolute convergence of the lattice coefficient sum follows from the explicit local majorant above. The limiting errors vanish for $\nu<k$ ; the separate pair argument has error $O(Q^{-1+1/k+\varepsilon}\log^2Q)$ after normalization, which tends to zero for every $k\ge2$ .

03Novelty0 reported findingsNo novelty concerns found

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Paper: arXiv:2507.00228v3
Authors listed: Bittu Chahal, Tapas Chatterjee, Sneha Chaubey
Audit date: October 24, 2026

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Distribution of Farey fractions with kkk-free denominators

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The asserted uniform Weyl bound fails at a common denominator

The constant term is wrong when the Farey set excludes one

The displayed pair density omits the modulus factor

Non-k-free coefficient tuples are also included incorrectly

The global distribution conclusions have verified arguments

The lattice method supports the existence assertions

The decay cannot be made uniform over all divisor sizes

The sawtooth constant depends on whether one is included

The harmonic progression main term has the wrong normalization and omits constants

The two local-density errors propagate into the central formula

The bounds assigned to the two coefficient indices are interchanged

The discrepancy requires a nonempty set

The valid ingredients and limiting ranges are checked separately

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Distribution of Farey fractions with $k$ -free denominators