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On the Distribution of Farey Fractions and Hyperbolic Lattice Points

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On the Distribution of Farey Fractions and Hyperbolic Lattice Points On the distribution of Farey fractions and hyperbolic lattice points Martin N. Huxley and Anatoly A. Zhigljavsky Cardi® University Abstract We derive an asymptotic formula for the number of pairs of con- secutive fractions a0=q0 and a=q in the Farey sequence of order Q such that a=q; q=Q; and (Q ¡ q0)=q) lie each in prescribed subintervals of the interval [0; 1]. We deduce the leading term in the asymptotic for- mula for 'the hyperbolic lattice point problem' for the modular group PSL(2; Z), the number of images of a given point under the action of the group in a given circle in the hyperbolic plane. Keywords: Farey fractions, hyperbolic geometry, lattice point problem, Selberg trace formula 1991 AMS Subject Classi¯cation: 11 B57, 11 F72. 1 Introduction The Farey sequence of FQ of order Q is the ¯nite set of rational numbers a=q with 1 · q · Q, 0 · a · q ¡ 1 and the highest common factor (a; q) = 1. The extended Farey sequence of order Q is de¯ned by dropping the condition 0 · a · q ¡ 1. It consists of all translates of the Farey sequence by integers. In both cases the Farey sequence is ordered by the relation x < y on real numbers. However to calculate one uses the lexicographic ordering, ¯rstly by q, and secondly by a. Some well-known properties are that if a0=q0 and a=q 1 are consecutive in FQ, then µ ¶ a a0 aq0 ¡ a0q = 1; 2 SL(2; Z) (1) q q0 where SL(2; Z) is the group of integer matrices of determinant one. In FQ the fraction a0=q0 determines its successor a=q uniquely through (1) and the restrictions q0 · Q, q + q0 > Q. Each pair of denominators q; q0 with high- est common factor (q; q0) = 1 and with q + q0 > Q occurs exactly once as 0 0 denominators of consecutive fractions a =q and a=q in FQ. So a pair of consecutive denominators in FQ corresponds to a primitive integer vector in the plane. Proposition 1 follows immediately by MÄobius inversion. Proposition 1. Let γ1; γ2; ±1; ±2 be real numbers with 0 · γ1 < γ2 · 1; 0 · ±1 < ±2 · 1 and let Q ¸ 2. Then the number of pairs of consecutive fractions a0=q0, a=q in FQ with q Q ¡ q0 ± < · ± ; γ < · γ (2) 1 Q 2 1 q 2 is 3 (γ ¡ γ )(±2 ¡ ±2)Q2 + O(Q log Q) (3) ¼2 2 1 2 1 uniformly in γ1; γ2; ±1; ±2. Proposition 1, or results of the same type, has been applied in questions of uniform distribution, see Kargaev and Zhigljavsky [4], Zhigljavsky and Aliev [11]. It is natural to consider also the position of a=q in the interval [0; 1). Theorem 1. Let ¯1; ¯2; γ1; γ2; ±1; ±2 be real numbers with 0 · ¯1 < ¯2 · 1; 0 · γ1 < γ2 · 1; 0 · ±1 < ±2 · 1; and let Q ¸ 2. Then the number of pairs of consecutive fractions a0=q0, a=q in FQ with a q Q ¡ q0 ¯ < · ¯ ; ± < · ± ; γ < · γ (4) 1 q 2 1 Q 2 1 q 2 2 is 3 (¯ ¡ ¯ )(γ ¡ γ )(±2 ¡ ±2)Q2 + O(Q3=2 log Q) : (5) ¼2 2 1 2 1 2 1 The matrix group SL(2; Z) acts as rigid motions in hyperbolic geometry. The action is faithful when we factor out §I, where I is the identity matrix. The quotient is called the modular group PSL(2; Z): Theorem 2. Let z and z0 be points in the hyperbolic plane, realised as the complex upper half-plane, with the modular group ¡ acting by z ¡! (az + b)=(cz + d). Then the number of images of z under ¡ in a circle with centre z0, radius r, in the hyperbolic metric, counted according to multiplicity, is asymptotically ³ ´ r (3+θ)r +²r 3e + O e 4 as r ! 1, for any ² > 0, where θ is an exponent such that the Kloosterman sums K(a; b; q) with highest common factor (a; b; q) = 1 satisfy K(a; b; q) = O(qθ+²) (6) for any ² > 0, as q ! 1. ¡ ¢ 3 The asymptotic formula of Theorem 2, with a better error term O exp( 4 r) , is a well-known consequence of the Selberg trace formula, see Patterson [5], and the proof of the trace formula is based on the image-counting function, see Hejhal [2]. A technical di±culty is to construct the spectral Eisenstein series. For the modular group the Eisenstein series is essentially an Epstein zeta function, corresponding to counting primitive lattice points inside el- lipses on the Euclidean plane. We sketch the proof of Theorem 2, which shows that the hyperbolic lattice point counting can also be done number theoretically. For the exponent θ in Theorem 2, we note that the estimate (6) 2 with θ = 3 can be proved quite simply, see Davenport [1] and Salie [6, 7]. Of 1 course Weil's bound [10] with θ = 2 is more di±cult than the Selberg trace 1 formula; the exponent θ = 2 can also be obtained by Stepanov's method, see Schmidt [8], akin to transcendence theory. 2 Proof of Theorem 1. We recall standard number-theoretical notation. We use (m; n) for the high- est common factor of m and n, d(n);!(n);Á(n); ¹(n) for the number of di- 3 visors of n, the number of distinct prime factors of n, the number of residue classes a mod n with (a; n) = 1 (Euler's function), and the MÄobiusfunction. For real t, e(t) = exp(2¼it) is the complex exponential function, [t] is the integer part of t and ½(t) = [t] ¡ t + 1 is the row-of-teeth function. In sums P 2 ¤ modulo q, ¤ means that the variable (a say) has (a; q) = 1, anda ¹ denotes the residue class d mod q with ad ´ 1, which exists when (a; q) = 1. The Kloosterman sum is µ ¶ X¤ ga + ha¹ K(g; h; q) = e : q a mod q 0 0 0 We note that if a =q , a=q are consecutive in FQ, then by (1) q ´ a¹ (mod q). For ¯xed q, the number of solutions of a Q ¡ a¹ ¯ < · ¯ ; γ < · γ (7) 1 q 2 1 q 2 is µ· ¸ · ¸¶µ· ¸ · ¸¶ X¤ a a Q ¡ a¹ Q ¡ a¹ ¯ ¡ ¡ ¯ ¡ γ ¡ ¡ γ ¡ : (8) 2 q 1 q 2 q 1 q a mod q We have · ¸ · ¸ µ ¶ µ ¶ a a a a ¯ ¡ ¡ ¯ ¡ = ¯ ¡ ¯ + ½ ¯ ¡ ¡ ½ ¯ ¡ (9) 2 q 1 q 2 1 2 q 1 q and similarly for the second factor. The main term is (¯2 ¡ ¯1)(γ2 ¡ γ1)Á(q) : Since X X Á(q) = ¹(d)e ±1Q<q·±2Q d;e: ± Q<de·± Q 1 2 µ µ ¶¶ X ±2Q2 ±2Q2 Q = ¹(d) 2 ¡ 1 + O 2d2 2d2 d d·Q µ µ ¶¶ 1 6 1 = (±2 ¡ ±2)Q2 + O + O(Q log Q) 2 2 1 ¼2 Q 3 = (±2 ¡ ±2)Q2 + O(Q log Q) (10) ¼2 2 1 4 we obtain the main term in Theorem 1. When we substitute (9) and its analogue involving γ1 and γ2 into (8), one cross-product term is µ ¶ µ µ ¶ µ ¶¶ X¤ a a γ ¡ γ ½ ¯ ¡ ¡ ½ ¯ ¡ 2 1 2 q 1 q a mod q µ ¶ µ µ ¶ µ ¶¶ X X a a = γ ¡ γ ¹(d) ½ ¯ ¡ ¡ ½ ¯ ¡ 2 1 2 q 1 q djq a mod q a´0 (mod d) µ ¶ µ µ ¶ µ ¶¶ X X b b = γ ¡ γ ¹(d) ½ ¯ ¡ ¡ ½ ¯ ¡ 2 1 2 q=d 1 q=d djq b mod q=d µ ¶ µ µ ¶ µ ¶¶ X q¯ q¯ = γ ¡ γ ¹(d) ½ 2 ¡ ½ 1 = O(d(q)) ; 2 1 d d djq where we have used the functional equations for ¹(d), ½ X 1 if d = 1; ¹(d) = 0 if d ¸ 2; djn and for ½(t), µ ¶ X b ½ t + = ½(nt) : n b mod n We have Q X X X X Q d(q) · 1 · · Q(log Q + 1) · 3Q log Q; d γ1Q<q·γ2Q q=1 djq d·Q so the cross-product term gives an error term of the same size as in (10). Similarly, the other cross-product term is µ µ ¶¶ µ ¶¶ X qγ ¡ Q qγ ¡ Q (¯ ¡ ¯ ) ¹(d) ½ 2 ¡ ½ 1 = O (Q log Q) : 2 1 d d djq The main error terms are four terms of the type µ ¶ µ ¶ X¤ a a¹ § ½ ¯ ¡ ½ ® ¡ (11) q q a mod q 5 where ¯ = ¯1 or ¯2 and ® = γ1 ¡ Q=q or γ2 ¡ Q=q. For each integer H ¸ 2 there are approximations XH XH b(h)e(ht) · ½(t) · B(h)e(ht) (12) h=¡H h=¡H (see Vaaler [9], section 8 or Huxley [3], section 5.3), with µ ¶ 1 b(0);B(0) = O ; H and for h 6= 0 µ ¶ 1 b(h);B(h) = O : jhj We use these to estimate the sum (11) from above and below. So we estimate H µ ¶ H µ ¶ X¤ X ag X ah¹ c(g)e g¯ ¡ C(h)e h® ¡ q q a mod q g=¡H h=¡H XH XH = c(g)e(g¯) C(h)e(h®)K(¡g; ¡h; q) ; (13) g=¡H h=¡H where c(n);C(n) are each either the set of coe±cients b(n) or B(n), and K(¡g; ¡h; q) is a Kloosterman sum, equal to K(g; h; q) by symmetry. Let e = (g; h; q) be the highest common factor of g; h; and q. The well- known bound p p jK(g; h; q)j · 2!(q=e) eq · d(q) eq; incorporates the theorem of Weil [10] and earlier work of Salie [6] .
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