Parabolas Infiltrating the Ford Circles

Bull. Math. Soc. Sci. Math. Roumanie Tome 59(107) No. 2, 2016, 175–185

Parabolas inﬁltrating the Ford circles by S. Corinne Hutchinson and Alexandru Zaharescu

Abstract We deﬁne and study a new family of parabolas in connection with Ford circles and discover some interesting properties that they have. Using these properties, we provide an application of such a family. This application, which establishes an asymptotic estimate for the L2 norm of a set of parabolas, proves useful in connection with recent work done by Kunik.

Key Words: Ford circles, Parabolas, Farey fractions. 2010 Mathematics Subject Classiﬁcation: Primary 11B57; Se- condary 11B99.

1 Introduction

In studying the set of fractions in 1938, L.R. Ford discovered and defined a new geometric object known as a Ford circle. Such a circle, as defined by Ford [7], is tangent to the x-axis at a given point with rational coordinates (a/q, 0), with a/q in lowest terms, and centered at (a/q, 1/2q2). For each such rational coordinate, a circle of such radius exists. Ford proved the following theorem in regards to these circles: The representative circles of two distinct fractions are either tangent or wholly external to one another. In the case that two circles are tangent to one another, Ford called the representative fractions adjacent. Several mathematicians, including Ford himself, have made connections between these circles and the set of Farey fractions. For some recent results on the distribution of Ford circles, the reader is referred to [1], [4]. For additional properties and results regarding Farey fractions refer to [2], [3], [5], and those references therein. A natural continuation of study, and that which the authors will discuss in this paper, is the discussion of the relationship between the rational numbers and other geometric objects, and connections that might be made between these 176 S. Corinne Hutchinson and Alexandru Zaharescu objects and Ford circles. In this spirit we define, at each rational a/q in reduced terms, a parabola in the following way:

q2 a2 P : y = x − . (1.1) a/q 2 q

These parabolas, as we will see below, exhibit many interesting geometric properties, as well as a close relationship with Ford circles. First, we ﬁx several families of objects which will be important in what follows. Namely, the family of all such parabolas as deﬁned in (1.1), the family of Ford circles, and the family of centers of such Ford circles: P = Pa/q : a/q ∈ Q C = Ca/q : a/q ∈ Q O = Oa/q : a/q ∈ Q . In Figure 1 these three families are shown for certain rational numbers in the interval [0, 1].

Figure 1: Some elements of the families P, C, and O.

In the following theorem, we collect some remarkable, yet easy to prove, geometric properties of these parabolas in connection with Ford circles.

Theorem 1. Given rational numbers r1 = a1/q1, r2 = a2/q2 in lowest terms,

Cr1 and Cr2 of diﬀerent radii, we have the following:

(i) The center Or2 lies on the parabola Pr1 if and only if the circles Cr1 and

Cr2 are tangent.

(ii) If the circles Cr1 and Cr2 are tangent, then the parabolas Pr1 and Pr2 intersect at exactly two points which are the centers of two Ford circles. Parabolas and Ford circles 177

(iii) If Cr3 and Cr4 are such that Cr1 is tangent to both, Cr2 is tangent to both,

and Cr1 and Cr2 are disjoint, then the parabolas Pr1 and Pr2 intersect at

two centers of Ford circles, namely Or3 and Or4 .

(iv) Reciprocity: The center Or2 lies on the parabola Pr1 if and only if Or1

lies on Pr2 . It is clear from the theorem that there is a distinct connection between the tangency of Ford circles and the points which lie on a given parabola as deﬁned. It may be interesting to study the points of intersection which are not centers of circles to see what statements might be made about these points. We ﬁrst prove the above theorem, and then present an application of this family of parabolas.

2 Proof of theorem 1

We ﬁrst prove (i). Given two circles, Cr1 and Cr2 , with r1 = a1/q1 and r2 = a2/q2, 2 2 then the radii of these circles are 1/2q1 and 1/2q2, respectively. Therefore, the circles are tangent if and only if the distance between their centers equals the 2 2 sum 1/2q1 + 1/2q2. This is true if and only if 2 2 2 a2 a1 1 1 1 1 − + 2 − 2 = 2 + 2 , (2.1) q2 q1 2q2 2q1 2q2 2q1 which further simpliﬁes to

a2 a1 1 − = . (2.2) q2 q1 q1q2

2 Next, the center Or2 = (a2/q2, 1/2q2) lies on the parabola Pr1 if and only if

2 2 1 q1 a2 a1 2 = − , (2.3) 2q2 2 q2 q1 which reduces to

1 a2 a1 = − . (2.4) q1q2 q2 q1

Therefore the center Or2 lies on the parabola Pr1 if and only if the circles Cr1 and Cr2 , are tangent. This completes the proof of (i). Similarly, the center OR1 lies on the parabola Pr2 if and only if Cr1 and Cr2 are tangent. Hence the center

Or1 lies on Pr2 if and only if Or2 lies on Pr1 . This proves (iv). For (ii), we provide a geometric proof. Refer to Figure 2 for the proof. Given two Ford circles Cr1 and Cr2 of diﬀerent radii, which are tangent to one another, by Ford’s theory we know that there are exactly two Ford circles, Cr3 and Cr4 , which are tangent to Cr1 and Cr2 . By (i), we know that Pr1 and Pr2 pass through both centers Or3 and Or4 . Since Pr1 and Pr2 cannot intersect at more than two points, it follows that their intersection consists exactly of Or3 and Or4 . This 178 S. Corinne Hutchinson and Alexandru Zaharescu completes the proof of (ii). The proof of (iii) follows from (i) in the same way as the proof of (ii) (refer to Figure 3). This completes the proof of Theorem 1.

Figure 2: The parabolas Pr1 , Pr2 , centers Or3 , Or4 , and representative circles of the rational numbers r1, r2, r3, and r4 as described in Theorem 1 (ii).

Figure 3: The parabolas Pr1 , Pr2 , centers Or3 , Or4 , and representative circles of the rational numbers r1, r2, r3, and r4 as described in Theorem 1 (iii). Parabolas and Ford circles 179

3 The set of parabolas associated to a set of Ford circles

A natural continuation of this study of parabolas is to consider the area given under the arcs of a certain set of parabolas. We then associate a set of parabolas with a finite set of Ford circles. To proceed, we fix a positive integer Q. Then we consider the finitely many Ford circles with centers on or above the horizontal 2 line y = 1/2Q . Let FQ be the ordered set of Farey fractions of order Q and let N = N(Q) be its cardinality. One knows that

3 N(Q) = Q2 + O(Q log Q). π2

We now define a real valued function FQ on the interval [0, 1] as follows. Let 0 0 a/q and a /q be consecutive fractions in FQ. From basic properties of Farey 0 0 fractions we know that the fraction θ = (a + a )/(q + q ) does not belong to FQ, and it is the first fraction that would be inserted between a/q and a0/q0 if one would allow Q to increase. By Ford’s theory, the circle Cθ is tangent to both Ca/q and Ca0/q0 . By its definition, Cθ is also tangent to the x-axis at the point 0 0 Tθ = (θ, 0). We define FQ on the interval [a/q, a /q ] in such a way that its graph over this interval coincides with the arc of the parabola Pθ lying above the same interval [a/q, a0/q0]. We know from Theorem 1 (i) that the end points of this arc are exactly the centers of the Ford circles Ca/q and Ca0/q0 . This shows that the function FQ is well defined and continuous on the entire interval [0, 1]. 0 0 On each subinterval of the form [a/q, a /q ], FQ is given explicitly by

(q + q0)2 a + a0 2 F (t) = t − . (3.1) Q 2 q + q0

In Figure 4, we show the Ford circles and the parabolas described above in the case Q = 5. Thus in this case the graph of FQ is a union of parabolas joining those centers of Ford circles that lie on or above the line y = 1/(2 · 52) = 1/50. Our next goal is to estimate the area under these parabolas, that is,

Z 1 Area(Q) = FQ(t) dt . 0

For example, in the particular case shown in Figure 4, the area under the parabolas equals Area(5) = 0.0675925. In Table 1 one sees how Area(Q) changes as Q increases.

Table 1: The size of Area(Q) and Q Area(Q) for selected values of Q. 180 S. Corinne Hutchinson and Alexandru Zaharescu

Figure 4: The set of Ford circles and parabolas with Q = 5.

QN(Q) Area(Q) Q · Area(Q) 2 3 0.1250000000 0.2500000000 10 33 0.0390560699 0.3905606996 100 3045 0.0044701813 0.4470181307 500 76117 0.0009078709 0.4539354502 1000 304193 0.0004549847 0.4549847094 5000 7600459 0.0000911752 0.4558762747 10000 30397487 0.0000456006 0.4560057781 20000 121590397 0.0000228035 0.4560703836 30000 273571775 0.0000152031 0.4560945837 40000 486345717 0.0000114026 0.4561061307 50000 759924265 0.0000091222 0.4561127077

Since the areas appear to decay like 1/Q, in the last column we also included the quantity Q Area(Q). Based on the data from the table it is natural to conjecture that the sequence Q Area(Q) Q≥1 converges to a certain constant, 0.456 ... The following theorem shows that this is indeed true. Theorem 2. For all positive integers Q,

ζ(2) 1 log Q Area(Q) = · + O , 3ζ(3) Q Q2 where ζ(s) denotes the Riemann zeta function. In particular, Theorem 2 conﬁrms the above conjecture, as lim Q Area(Q) = ζ(2)/3ζ(3) = 0.45614 ... Q→∞ Parabolas and Ford circles 181

It would be interesting to further study the oscillatory behavior of the function Q → Q Area(Q), shown in Figure 5.

Figure 5: The behavior of the function Q → Q Area(Q).

4 Proof of theorem 2

Recall that between two consecutive Farey fractions α = a/q and β = a0/q0, 2 0 0 FQ(t) has the form FQ(t) = A(t − θ) , where θ = (a + a )/(q + q ) is the point of tangency to the x-axis and A = (q + q0)2/2. Thus

Z β 3 β A(t − θ) FQ(t) dt = . (4.1) α 3 α To simplify the integral, notice that

a0 a + a0 1 β − θ = − = , q0 q + q0 q0(q + q0) a a + a0 1 α − θ = − = − , q q + q0 q(q + q0) since by a fundamental property of Farey fractions, a0q − aq0 = 1. We ﬁnd that the integral in (4.1) reduces to

Z β 1 1 1 F (t) dt = 3 0 + 03 0 . α 6 q (q + q ) q (q + q )

Taking into account the symmetry of the Farey sequence with respect to 1/2, or in other words taking into account that for each pair of consecutive Farey fractions a/q and a0/q0 there is another pair having the same denominators q, q0, in reverse order, namely 1 − a0/q0 and 1 − a/q, one sees that

1 X 1 Area(Q) = , (4.2) 3 q3(q + q0) 182 S. Corinne Hutchinson and Alexandru Zaharescu where the sum is over all pairs (q, q0) of consecutive denominators of the sequence of Farey fractions of order Q. From basic properties of Farey fractions we know that this set of pairs of consecutive denominators coincides with the set of pairs (q, q0) of integer numbers for which 1 ≤ q, q0 ≤ Q, gcd(q, q0) = 1 and q + q0 > Q. Thus we may rewrite (4.2) as

1 X 1 1 Area(Q) = := S(Q) . (4.3) 3 q3(q + q0) 3 1≤q,q0≤Q gcd(q,q0)=1 q+q0>Q

1 For convenience, we introduce the function f(x, y) = x3(x+y) and the trian- gular region Ω = Ω(Q), which is deﬁned by the inequalities 1 ≤ x, y ≤ Q and x + y > Q. Then the sum in (4.3) can be written as X S(Q) = f(x, y) . (x,y)∈ Ω gcd(x,y)=1

By M¨obiussummation, X X S(Q) = f(x, y) µ(d) (x,y)∈ Ω d| gcd(x,y) X X (4.4) = µ(d) f(x, y) . 1≤d≤Q (x,y)∈ Ω d|x, d|y

We continue the evaluation in (4.4) by changing the variables x = dm and y = dn, to obtain X X S(Q) = µ(d) f(dm, dn) 1≤d≤Q 1 (m,n)∈ d Ω (4.5) X µ(d) X 1 X µ(d) = := S (d, Q) . d4 m3(m + n) d4 1 d≤Q 1 d≤Q (m,n)∈ d Ω

The inner sum S1(d, Q) can be written as

X 1 X 1 S (d, Q) = 1 m3 m + n 1≤m≤Q/d Q/d−m

Here we introduce a variable s to take into account the size of r near Q/d. We Parabolas and Ford circles 183 obtain m X 1 X 1 S1(d, Q) = m3 h Q i 1≤m≤Q/d s=1 d + s m X 1 X 1 = m3 Q n Q o 1≤m≤Q/d s=1 d − d + s m d X 1 X sd = 1 + O Q m3 Q 1≤m≤Q/d s=1 d X 1 d2 log Q/d = + O . Q m2 Q2 1≤m≤Q/d The completion of the sum over all positive integers m, yields

∞ d X 1 d X 1 d2 log Q/d S (d, Q) = − + O 1 Q m2 Q m2 Q2 m=1 m>Q/d (4.6) dζ(2) d2 log Q/d = + O . Q Q2 Inserting this estimate into (4.5), we ﬁnd that

X µ(d) dζ(2) d2 log Q/d ζ(2) 1 log Q S(Q) = + O = · + O . d4 Q Q2 ζ(3) Q Q2 d≤Q Hence, in view of (4.3), the proof of the theorem is completed.

5 Application to Kunik’s function

∗ In a recent paper, Kunik [8] deﬁnes a function t 7→ ΨQ(t), for each positive integer aj−1 Q, as follows. Between any two consecutive Farey fractions in FQ, say and qj−1 aj , j = 1,...,N(Q) − 1, the function Ψ∗ (t) is deﬁned by: qj Q ∗ qj + qj−1 aj + aj−1 aj−1 aj ΨQ(t) := − t − , for ≤ t < . 2 qj + qj−1 qj−1 qj

∗ The function ΨQ(t) is a step function, being linear between subsequent Farey aj fractions in FQ and having a jump of height 1/qj at each Farey fraction . Kunik qj established a number of interesting results, including the following L2-estimate: Z 1 ∗ 2 ∗ 2 log Q ||ΨQ||2 = ΨQ(t) dt = O . 0 Q 184 S. Corinne Hutchinson and Alexandru Zaharescu

We notice a connection between the above function FQ(t) and Kunik’s func- ∗ 2 tion ΨQ(t). More precisely, we restate an improved L -estimate of Kunik’s function in the following theorem.

Theorem 3. For positive integers Q ≥ 1,

Z 1 ∗ 2 1 ζ(2) 1 log Q ΨQ(t) dt = Area(Q) = · + O 2 , 0 2 6ζ(3) Q Q where ζ(s) denotes the Riemann zeta function. The proof of this theorem is as follows. We ﬁrst notice the connection between ∗ FQ(t) and the function ΨQ(t), which says that

∗ 2 FQ(t) = 2ΨQ(t) , for all t ∈ [0, 1] and Q ≥ 1.

From here, it is straightforward to see that by Theorem 2, one may replace the original L2-estimate by the one given in the theorem. Thus the proof of Theorem 3 is complete.

References

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Received: 27.04.2015 Revised: 06.06.2015 Accepted: 25.06.2015 Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA E-mail: [email protected]

Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania and University of Illinois, Department of Mathematics, 1409 West Green Street, Urbana, IL 61801, USA E-mail: [email protected]