K-MOMENTS of DISTANCES BETWEEN CENTERS of FORD CIRCLES 1. Introduction Introduced in 1938 by Lester R. Ford

k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES

SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU

Abstract. In this paper, we investigate a problem on the distribution of Ford circles, which concerns moments of distances between centers of these circles that lie above a given horizontal line.

1. Introduction Introduced in 1938 by Lester R. Ford [8], a Ford circle is a circle tangent to the x-axis at a given point with rational coordinates (p/q, 0) in reduced form, centered at (p/q, 1/(2q2)). Any two Ford circles are either disjoint or tangent to each other. In the present paper, we study a question concerning the distribution of Ford circles. For a ﬁxed interval I := [α, β] ⊆ [0, 1] with rational end points and for each large positive integer Q, we consider the set FI,Q consisting of Ford circles with centers lying between the 1 vertical lines x = α and x = β or possibly on the line x = β but not below the line y = 2Q2 . Note that these are the Ford circles that are tangent to the real axis at the rational points (a/q, 0) with a/q in the interval I = [α, β] and q ≤ Q. Let NI (Q) denote the number of

elements in FI,Q. The circles CQ,1,CQ,2, ··· ,CQ,NI (Q) in FI,Q are arranged in such a way that any two consecutive circles are tangent to each other. For each j in {1, 2, ··· ,NI (Q)}, denote the center of CQ,j by OQ,j and the radius of CQ,j by rQ,j. For any positive integer k, consider the k-moment

NI (Q)−1 1 X k M (Q) := (D(O ,O )) , (1.1) k,I |I| Q,j Q,j+1 j=1

where D(OQ,j,OQ,j+1) denotes the Euclidean distance between the centers OQ,j and OQ,j+1. For all large X, we consider the average

1 Z 2X Ak,I (X) := Mk,I (Q) dY, (1.2) X X where here and in what follows, Y denotes a real variable and the positive integer Q is a function of Y ; more precisely, Q is the integer part of Y . Although, as Q increases, the sequence of individual distances D(QQ,j,OQ,j+1) changes wildly as more and more circles of various sizes appear between any two given circles, the k-averages Ak,I (X) satisfy nice asymptotic formulas.

2010 Mathematics Subject Classiﬁcation. 11B57, 11M06. Key words and phrases. Ford circles, Farey Fractions, Riemann zeta function, Kloosterman sums. 1 2 SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU

Theorem 1.1. Fix an interval I := [α, β] ⊆ [0, 1] with α, β ∈ Q, and let Ak,I (X) be deﬁned as in (1.2). Then, for k = 1, 6 log2 X A (X) = log 4X + B (I) + O , (1.3) 1,I π2 1 X

where B1(I) is a constant depending only on the interval I.

We remark that when I = [0, 1], the constant B1(I) is given by γ − 1 ζ0(2) B ([0, 1]) = − , 1 ζ(2) ζ2(2) where ζ(s) denotes the Riemann zeta function and γ denotes Euler’s constant. In this case, we also obtain a better bound for the error term in (1.3), namely, 6 γ − 1 ζ0(2) 1 A1,[0,1](X) = log 4X + − + O 3/5 −1/5 , (1.4) π2 ζ(2) ζ2(2) Xec0(log X) (log log X)

where c0 > 0 is an absolute constant. Theorem 1.2. For I as in Theorem 1.1 and k = 2 in (1.2), 3 log X D (I) A (X) = B (I) + + 2 + O X−21/10+ , 2,I 2 π2 X2 X2 where B2(I) and D2(I) are constants depending only on the interval I. In particular, for I = [0, 1], one has ζ(3) 3 ζ0(2) 5 B ([0, 1]) = and D ([0, 1]) = γ − + − log 2 . 2 2ζ(4) 2 π2 ζ(2) 4 In this case also, we obtain a better bound for the error term, ζ(3) 3 log X 3 ζ0(2) 5 1 A (X) = + + γ − + − log 2 2,[0,1] 2ζ(4) π2 X2 π2 ζ(2) 4 X2 ! log5/3 X(log log X)1+ + O . (1.5) X3

Theorem 1.3. For I as in Theorem 1.1 and k ≥ 3 in (1.2), 1 A (X) = B (I) + O , k,I k k X2

where Bk(I) is a constant depending on k and the interval I. In particular, for the full interval I = [0, 1], one has ζ(2k − 1) B ([0, 1]) = . k 2k−1ζ(2k) In this case, we obtain a second order term and a better bound for the error term, ζ(2k − 1) kζ(2k − 3) 1 1 A (X) = + + O . (1.6) k,[0,1] 2k−1ζ(2k) 2kζ(2k − 2) X2 k X3 It would be interesting to investigate similar questions for Apollonian circle packings. k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 3

2. General Setup

In this section, we ﬁx a positive integer k and express the k-th moment Mk,I in terms of the Euler-phi function and the Mobius function. Next, we rewrite Ak,I as an integral involving the Riemann zeta function, and then shift the path of integration based on the Vinogradov-Korobov zero free region. To proceed, we ﬁrst review some facts about Farey fractions. Given a positive integer Q, by a Farey fraction of order Q, we mean a rational number in reduced form in the interval [0, 1] with denominator at most Q. We denote by FQ the sequence of Farey fractions of order Q, arranged in order of increasing size. Two Farey fractions a/b < c/d in FQ are neighbours if and only if bc − ad = 1 and b + d > Q. The Farey sequence FQ is in bijection with the set of Ford circles tangent to the real line at points in 1 the interval [0, 1] and radius at least 2Q2 . Therefore, the distance between the centers OQ,j and OQ,j+1 of two consecutive Ford circles CQ,j and CQ,j+1 is given by

1 1 D(QQ,j,OQ,j+1) = 2 + 2 , (2.1) 2qj 2qj+1

where aj/qj and aj+1/qj+1 are neighbours in FQ and correspond to a pair of Ford circles tangent to the x-axis at x = aj/qj and x = aj+1/qj+1 respectively. Thus, NI (Q) is same as the number of Farey fractions of order Q inside the interval I = [α, β]. For two neighbouring Farey fractions a0/q0 < a/q of order Q in the interval I, we note that a0 ≡ −q¯ (mod q0), since q0a − qa0 = 1. The notation x¯ (mod n) is used for the multiplicative inverse of x (mod n) in the interval [1, n] for positive integers x and n with gcd(x, n) = 1. Thus, the conditions a0/q0 ∈ (α, β] and a/q ∈ (α, β] are equivalent to

q¯ ∈ [q0 − q0β, q0 − q0α), and q¯0 ∈ (qα, qβ], (2.2)

respectively. Questions on the distribution of Farey fractions have been studied extensively, see for example [1], [2], [4], [9] and the references therein. For a ﬁxed k and the interval I, from (1.1) and (2.1), one has

NI (Q)−1 k X 1 1 |I|M (Q) = + k,I 2q2 2q2 j=1 j j+1

NI (Q) NI (Q)−1 k−1 !2 1 X 1 1 1 1 X X k 1 = + − + 2k−1 q2k 2kq2k 2kq2k 2k i qi qk−i j=2 j 1 NI (Q) j=1 i=1 j j+1

k−1 NI (Q)−1 ! 1 X 1 X 1 X k X 1 1 1 = 1 + + − 2k−1 q2k 2k i q2iq2k−2i 2kq2k 2kq2k 1≤q≤Q αq

where in the last inequality, without loss of generality, we assume that for large Q, the endpoints of the interval I = [α, β] are Farey fractions of order Q since α, β ∈ Q. This implies 4 SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU

Rk(I) in (2.3) is a constant depending only on k and end-points of the interval I. Therefore,

Z 2X 1 0 |I|Ak,I = (Sk + Sk + Rk(I)) dY X X Z 2X Z X Z 2X 1 1 1 0 = Sk dY − Sk dY + Sk dY + Rk(I). (2.4) X 1 X 1 X X

Consider,

1 Z X 1 Z X X 1 X S dY = 1 dY X k 2k−1X q2k 1 1 1≤q≤Q αq

=: Sk,1 − Sk,2 + Sk,3. (2.5)

We ﬁrst estimate the sums Sk,2 and Sk,3. Since for x ≥ 1, and a ≥ 2,

X 1 = O x1−a , na n≥x k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 5

one has   1 X µ(d) X {βm} − {αm} X {βm} − {αm} S = − k,2 2k−1 d2k  m2k m2k  1≤d≤X m≥1 m>X/d   C2k,I X µ(d) X |µ(d)| X |{βm} − {αm}| = + O 2k−1 d2k  d2k m2k  1≤d≤X 1≤d≤X m>X/d C log X = 2k,I + O , (2.6) 2k−1ζ(2k) X2k−1 where for a natural number j, we denote X {βm} − {αm} C := . j,I mj m≥1 Similarly for k ≥ 2,   1 X µ(d) X {βm} − {αm} X {βm} − {αm} S = − k,3 2k−1X d2k−1  m2k−1 m2k−1  1≤d≤X m≥1 m>X/d   C2k−1,I 1 X µ(d) 1 X |µ(d)| X |{βm} − {αm}| = + O 2k−1 X d2k−1 X d2k−1 m2k−1  1≤d≤X 1≤d≤X m>X/d C 1 log X = 2k−1,I + O . (2.7) 2k−1ζ(2k − 1) X X2k−1

In a similar fashion one estimates Sk,3 for k = 1,

1 X µ(d) X {βm} − {αm} |S1,3| = X d m 1≤d≤X m 2−2k. For a complex number s, we write s = σ+it. By Perron’s formula ([12, page 130]), for c > 0, 1 X φ(q) 1 Z c+i∞ Xsζ(s + 2k − 1) (X − q) = ds. (2.9) X q2k 2πi s(s + 1)ζ(s + 2k) q≤X c−i∞ 6 SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU a Fix T, U > 0 such that 2 ≤ T ≤ X, X2 ≤ U ≤ X2k and c = for some absolute log X constant a > 0. Let A d = −2k + 1 − , (log 2T )2/3(log log 2T )1/3 where A will be a suitably chosen absolute constant. In order to evaluate the above integral, we modify the path of integration from c − iU to c + iU along the line segments lj, 1 ≤ j ≤ 9 described below. We let l1 be the half line from c + iU to c + i∞, l2 be the line segment from −2k + 1 + iU to c+iU, l3 be the line segment from −2k +1+iT to −2k +1+iU, l4 be the line segment from d + iT to −2k + 1 + iT , l5 be the line segment from d − iT to d + iT, l6 be the line segment from −2k + 1 − iT to d − iT , l7 be the line segment from −2k + 1 − iU to −2k + 1 − iT, l8 be the line segment with endpoints −2k + 1 − iU and c − iU, and lastly let l9 be the half line from c − i∞ to c − iU. The main contribution on the right side of (2.9) comes from the residues at the poles of the function Xsζ(s + 2k − 1) f (s) := , k s(s + 1)ζ(s + 2k) encountered when we modiﬁed the path of integration. By the residue theorem,

9 1 Z c+i∞ X X f (s) ds = Res(f (s)) + J , (2.10) 2πi k k m c−i∞ m=1 X where Ji is the integral of fk(s) along li. Here the sum Res(fk(s)) is taken over all the

poles of fk(s) inside the region bounded by segments l2, l3, . . . , l8 and the vertical segment joining c − iU and c + iU. To estimate the integrals J1,...,J9, we use standard bounds for 1 ζ(s) and ([13, page 47]), ζ(s)  σ− 1  O t 2 log t , −1 ≤ σ ≤ 0,   1−σ ζ(σ + it) = O t 2 log t , 0 ≤ σ ≤ 1,   O (log t) , 1 ≤ σ ≤ 2,  O (1) , σ ≥ 2, and 1 O (log t) , 1 ≤ σ ≤ 2, = ζ(σ + it) O (1) , σ ≥ 2. We also use the Vinogradov-Koroborov zero free region ([14], [11]),

− 2 − 1 σ ≥ 1 − B(log t) 3 (log log t) 3 , where 1 2 1 = O (log t) 3 (log log t) 3 , ζ(s) and B is an absolute constant. k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 7

3. First Moment In this section we provide a proof of Theorem 1.1. For the ﬁrst moment, we set k = 1 in Section 2. Proof of Theorem 1.1. From (2.3), we note that ! X 1 X 1 1 |I|M = 1 + − + = S + R (I). 1,I q2 2q2 2q2 1 1 q≤Q αq

The sums S1,2 and S1,3 have already been estimated in (2.6) and (2.8) respectively. In order to estimate S1,1, we bound the integrals Jm in (2.10) as follows. One has Z ∞ |Xc+it||ζ(c + 1 + it)| |J1|, |J9| = O dt U |c + it||c + 1 + it||ζ(c + 2 + it)| Z ∞ c log t log U = O X 2 dt = O . U t U And, Z c |Xσ+iU ||ζ(1 + σ + iU)| |J2|, |J8| = O dσ −1 |σ + iU||σ + 1 + iU||ζ(2 + σ + iU)| (log U)2 Z 0 X σ log U Z c log2 U √ σ = O 2 dσ + 2 X dσ = O 2 . U −1 U U 0 U Next, Z U −1+it Z U |X ||ζ(it)| −1 log t |J3|, |J7| = O dt = O X 3/2 dt T | − 1 + it||it||ζ(1 + it)| T t log2 T = O √ . X T Also, Z −1 |Xσ+iT ||ζ(1 + σ + iT )| |J4|, |J6| = O dσ d |σ + 1 + iT ||σ + iT ||ζ(σ + 2 + iT )| 2 1 ! 3 3 Z −1 (log T ) (log log T ) σ = O 2 X |ζ(1 + σ + iT )| dσ T d 5 1 σ ! 2 1 ! (log T ) 3 (log log T ) 3 Z −1 X (log T ) 3 (log log T ) 3 √ = O 2 dσ = O 3/2 . T d T XT 8 SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU

Lastly, Z T |Xd+it||ζ(1 + d + it)| |J5| = O dt −T |d + it||d + 1 + it||ζ(d + 2 + it)| Z T −1/2−d 5/3 1/3 ! d t (log(2 + |t|) (log log(3 + |t|)) d = O X 2 dt = O X . −T 1 + t ! c (log X)3/5 Collecting all the above estimates and setting U = X2 and T = exp 1 , one (log log X)1/5 obtains 1 S1,1 = |I| Res(f1(s)) + O 3/5 −1/5 , (3.3) Xec0(log X) (log log X)

where c0 and c1 are suitable positive absolute constants. Here, in the prescribed region, f1(s) has only one pole at s = 0 of order two with residue log X γ − 1 ζ0(2) Res(f (s)) = + − . 1 ζ(2) ζ(2) ζ2(2) From (2.6), (2.8), (3.1), (3.2), (3.3) and above, 6 γ − 1 ζ0(2) C R (I) log2 X A (X) = log 4X + − − 2,I + 1 + O . 1,I π2 ζ(2) ζ2(2) |I|ζ(2) |I| X

This concludes the proof of Theorem 1.1.

Remark. In the case of the full interval I = [0, 1], we observe that R1([0, 1]) = 0 in (2.3) and S1,2 = 0 = S1,3 in (2.5). Therefore, S1,1 in (3.2) is the only term which contributes to the average A1,[0,1] in (3.1) and we obtain 6 γ − 1 ζ0(2) 1 A1,[0,1](X) = log 4X + − + O 3/5 −1/5 , π2 ζ(2) ζ2(2) Xec0(log X) (log log X) as claimed in (1.4).

4. Higher Moments In this section, we prove Theorem 1.2 and Theorem 1.3. We ﬁrst estimate the integral 1 Z X Sk dY for k ≥ 2 in (2.4). From (2.5), X 1 1 Z X Sk dY = Sk,1 − Sk,2 + Sk,3. X 1

Estimates for Sk,2 and Sk,3 for k ≥ 2 have already been obtained in (2.6) and (2.7). For k ≥ 2, estimates for |I| X φ(q) S = (X − q) k,1 2k−1X q2k q≤X k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 9

can be obtained as before where we set U = X2k. In this case, the corresponding function fk(s) has poles at s = 0, s = −1 and s = 2 − 2k in the region described before. All these poles are simple and the sum of the residues of fk(s) at these poles is given by X ζ(2k − 1) ζ(2k − 2) 1 1 1 Res(f (s)) = − + . k ζ(2k) ζ(2k − 1) X (2k − 3)(2k − 2)ζ(2) X2k−2

One can estimate the line integrals Jm of the function f1(s) along li for 1 ≤ i ≤ 9 in (2.10) as before. In this case one has 1 Z c+i∞ Xsζ(s + 2k − 1) ζ(2k − 1) ζ(2k − 2) 1 1 ds = − + 2k−2 2πi c−i∞ s(s + 1)ζ(s + 2k) ζ(2k) ζ(2k − 1) X (2k − 3)(2k − 2)ζ(2)X 1 + O 3/5 −1/5 . X2k−1ec0(log X) (log log X) Therefore, from (2.9) and the above equation, we obtain |I|ζ(2k − 1) |I|ζ(2k − 2) 1 |I| S = − + k,1 2k−1ζ(2k) 2k−1ζ(2k − 1) X 2k−1(2k − 3)(2k − 2)ζ(2)X2k−2 1 + O 3/5 −1/5 . X2k−1ec0(log X) (log log X) From (2.5), (2.6), (2.7) and above, we derive Z 2X 2k−3 1 |I|ζ(2k − 1) C2k,I |I|(1 − 2 ) Sk dY = k−1 − k−1 + 3k−4 2k−2 X X 2 ζ(2k) 2 ζ(2k) 2 (2k − 3)(2k − 2)ζ(2)X log X + O . (4.1) X2k−1 In order to prove Theorem 1.2 and Theorem 1.3, it remains to estimate the remaining integral Z 2X 1 0 Sk dY X X for k ≥ 2 in (2.4). Proof of Theorem 1.2. For k = 2 in (2.3), we have

NI (Q)−1 2 X 1 1 |I|M (Q) = + 2,I 2q2 2q2 j=1 j j+1

NI (Q)−1 2 ! 1 X 1 X 1 X 1 1 1 = 4 1 + + − 4 + 4 2 q 2 qjqj+1 4q 4q q≤Q αq

where ! 12 ζ0(2) 1 log5/3 Q(log log Q)1+ S (Q) = log Q + γ − + + O . 0 π2Q2 ζ(2) 2 Q3 We remark in passing that the saving in the exponent above (from −2 to −21/10) was obtained by employing Weil type estimates ([7], [10], [15]) for Kloosterman sums. Next, we have Z 2X 0 1 0 3 log X 3 ζ (2) 3 C2,I 1 −21/10+ S2 dY = 2 2 + 2 γ − + − log 2 + 2 + O X . |I|X X π X π ζ(2) 2 2|I| X Combining (4.1), (4.2) and above, we conclude that |I|ζ(3) − C R (I) 3 log X 3 ζ0(2) 5 C 1 A = 4,I + 2 + + γ − + − log 2 + 2,I 2,I 2|I|ζ(4) |I| π2 X2 π2 ζ(2) 4 2|I| X2 −21/10+ + O X . This completes the proof of Theorem 1.2.

Remark. For the full interval I = [0, 1], observe that R2(I) = 0, and ! 6 ζ0(2) 1 log5/3 Q(log log Q)1+ S0 = log Q + γ − + + O . 2 π2Q2 ζ(2) 2 Q3 This along with (4.1) and (4.2) proves (1.5), ζ(3) 1 3 log X 3 ζ0(2) 5 1 A (X) = − + + γ − + − log 2 2,[0,1] 2ζ(4) 2 π2 X2 π2 ζ(2) 4 X2 ! log5/3 X(log log X)1+ + O . X3

Proof of Theorem 1.3. For k ≥ 3,

NI (Q)−1 k X 1 1 |I|M (Q) = + k,I 2q 2 2q 2 j=1 j j+1

k−1 NI (Q)−1 !2 ! X 1 X |I| 1 X k X 1 1 1 = + − + q2k 2k−1 2k i qi qk−i 2kq2k 2kq2k q≤Q αq

NI (Q)−1 X 1 S := . k,i 2i 2k−2i j=1 qj qj+1 k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 11

For any positive integer m, let Lm denote the set ( ) l ∈ N: l > m, Q − m < l ≤ Q, gcd(m, l) = 1, L := . m ¯l (mod m) ∈ (mα, mβ], m¯ (mod l) ∈ [l − lβ, l − lα)

Employing (2.2), we have

X X 1 X X 1 S = + . k,i q2ir2k−2i q2ir2k−2i 1≤r≤Q q∈Lr 1≤q≤Q r∈Lq

As noted earlier in Section 2, when q and r are denominators of neighbouring Farey fractions in FQ, then r + q > Q. Therefore, q > r implies q > Q/2 and for r > q, we have r > Q/2. Also, X X 1 = O (φ(r)) and 1 = O (φ(q)) .

q∈Lr r∈Lq

Using the above relations and the fact that for x ≥ 2 and a ≥ 3,

X φ(n) ζ(a − 1) = + O x2−a , (4.4) na ζ(a) 1≤n≤x

we obtain

2i 2k−2i 2 X 1 X 2 X 1 X S ≤ 1 + 1 k,i Q r2k−2i Q q2i r≤Q q∈Lr q≤Q r∈Lq ! ! 1 X 1 1 X 1 = O φ(r) + O φ(q) Q2i r2k−2i Q2k−2i q2i r≤Q q≤Q log Q log Q = O + O . Q2i Q2k−2i

Here on the far right side, the ﬁrst log Q may be replaced by 1 unless i = k − 1, and the second log Q may be replaced by 1 unless i = 1. Hence,

k−1 1 Z 2X 1 Z 2X 1 X k 1 S0 dY = S dY = O . X k X 2k i k,i X2 X X i=1

This combined with (4.1) and (4.3) yields

|I|ζ(2k − 1) − C R (I) 1 A = 2k,I + k + O for k ≥ 3, k,I |I|2k−1ζ(2k) |I| k X2 which completes the proof of Theorem 1.3. 12 SNEHA CHAUBEY, AMITA MALIK, AND ALEXANDRU ZAHARESCU

Remark. In the case of the full interval I = [0, 1], note that Rk([0, 1]) = 0, and X 1 S = k,i q2ir2k−2i 1≤q,r≤Q gcd(q,r)=1, q+r>Q X 1 X 1 X 1 = + + q2ir2k−2i q2ir2k−2i q2ir2k−2i 1≤q,r≤Q 1≤q,r≤Q 1≤q,r≤Q gcd(q,r)=1, gcd(q,r)=1, gcd(q,r)=1, rQ q+r>Q

=: Σ1,i + Σ2,i + Σ3,i.

First we estimate the sum Σ1,i for 1 ≤ i ≤ k − 2. Note that in this case r < Q/2, therefore 1 1 Q − q q > Q/2 since r + q > Q. Also, = 1 + O gives, q Q Q   X 1  X 1 Q − q  Σ1,i =  + O  r2k−2i  Q2i Q2i+1  1≤r

Similarly, for the sum Σ2,i for 2 ≤ i ≤ k − 1, ζ(2i − 1) 1 1 Σ = + O , 2,i ζ(2i) Q2k−2i Q2k−2i+1 and 1 log(Q/2) 1 Σ = + O . 2,1 ζ(2) Q2k−2 Q2k−2 k-MOMENTS OF DISTANCES BETWEEN CENTERS OF FORD CIRCLES 13

Lastly, for 1 ≤ i ≤ k − 1, 1 Σ = O . 3,i Q2k−2 Therefore, kζ(2k − 3) 1 1 S0 = + O . k 2k−1ζ(2k − 2) Q2 Q3 This combined with (4.3) gives (1.6), ζ(2k − 1) 1 kζ(2k − 3) 1 1 A = − + + O . k,[0,1] 2k−1ζ(2k) 2k−1 2kζ(2k − 2) X2 k X3 References [1] J. S. Athreya, Y. Cheung, A Poincaré section for horocycle ﬂow on the space of lattices, to appear in Int. Math. Res. Not. IMRN. [2] A. D. Badziahin, A. K. Haynes, A note on Farey fractions with denominators in arithmetic progressions, Acta Arith. 147 (2011), 205–215. [3] F. Boca, C. Cobeli and A. Zaharescu, A conjecture of R.R. Hall on Farey points, J. Reine Angew. Math. 535 (2001), 207–236. [4] F. Boca and A. Zaharescu, The correlations of Farey fractions, J. London Math. Soc. 72 (2005), 25–39. [5] C. Cobeli, M. Vâjâitu and A. Zaharescu, A density theorem on even Farey fractions, Rev. Roumaine Math. Pures Appl. 55 (2010), 447–481. [6] C. Cobeli and A. Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform. 5 (2003), 1–38. [7] T. Estermann, On Kloosterman’s sum, Mathematika 8 (1961), 83–86. [8] L. R. Ford, Fractions, Amer. Math. Monthly 45 (1938), 586–601. [9] A. K. Haynes, A note on Farey fractions with odd denominators, J. Number Theory 98 (2003), 89–104. [10] C. Hooley, An asymptotic formula in the theory of numbers, Proc. London Math. Soc. 7 (1957), 396–413. [11] N. M. Korobov, Estimates of trigonometric sums and their applications, Uspehi Mat. Nauk 13 (1958), 185–192. (Russian) [12] G. Tenenbaum, Introduction to analytic and probabilistic number theory, (Translated from the second French edition by C. B. Thomas), Cambridge Studies in Advanced Mathematics, 46, Cambridge Uni- versity Press, (Cambridge, 1995). [13] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed. (Edited and with a preface by D. R. Heath-Brown), The Clarendon Press, Oxford University Press (New York, 1986). [14] I. M. Vinogradov, A new estimate for ζ(1 + it), Izv. Akad. Nauk SSSR, Ser. Mat. 22 (1958), 161–164. (Russian) [15] A. Weil, On some exponential sums, Proc. Nat. Acad. U.S.A. 34 (1948), 204–207.

Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA. E-mail address: [email protected]