Rational approximations to π and some other numbers

Transcription

1 ACTA ARITHMETICA LXIII.4 (993) Rational approximations to π and some other numbers by Masayoshi Hata (Kyoto). Introduction. In 953 K. Mahler [2] gave a lower bound for rational approximations to π by showing that π p q q 42 for any integers p,q with q 2. He also indicated that the exponent 42 can be replaced by 30 when q is greater than some integer q 0. This result is based on the classical approximation formulae to the exponential and logarithmic functions due to Hermite. Our aim is to improve the knowledge of approximations by rational numbers of the classical constants of analysis such as π and π/ 3. In general, we say that a real number γ has an irrationality measure µ 2 provided that, for any ε > 0, there exists a positive integer q 0 (ε) satisfying γ p q q µ ε for any integers p,q with q q 0 (ε). Later the exponent 30 mentioned above was improved to 20 by M. Mignotte [3] by a refinement of Mahler s method, and to by G. V. Chudnovsky [3]. Indeed, Chudnovsky has succeeded in determining the exact asymptotic behaviour of the complex integral (.) 2πi C ( ) k n! e wz dz z(z )... (z n) as n tends to infinity. Recently the author [], modifying the above integral (.), obtained a fairly good irrationality measure for π. On the other hand, F. Beukers [2] gave an elegant irrationality proof of ζ(2) = π 2 /6 by considering the double (real) integral

2 336 M. Hata (.2) 0 0 L(x)( y) n dxdy, xy where L(x) = n! (xn ( x) n ) (n) is the Legendre polynomial. The integral (.2) gives an irrationality measure for π 2 and this has been pared down to by R. Dvornicich and C. Viola [9], to by E. A. Rukhadze [5], to by the author [0], and to by D. V. Chudnovsky and G. V. Chudnovsky [6]. (The measures and were announced without proofs.) Thus the last result due to Chudnovsky and Chudnovsky brings an irrationality measure for π. All the measures mentioned above are based on rational approximations to π and its powers, though a single system of approximation to π is desired. In this paper we consider the complex integral (.3) (F(a,a 2,a 3 ;z)) n dz z where Γ (.4) F(a,a 2,a 3 ;z) = (z a ) 2 (z a 2 ) 2 (z a 3 ) 2 z 3 with non-zero distinct complex numbers a,a 2 and a 3. By taking a =, a 2 = 2 and a 3 = + i, the integral (.3) enables us to obtain the following Theorem.. For any ε > 0, there exists a positive integer H 0 (ε) such that p + qπ + r log 2 H µ ε for any integers p,q,r with H max{ q, r } H 0 (ε), where the exponent µ is given by µ = 2log α π/ 3 2log α + 6 π/ 3 with α j = cos (θ 0 + 4jπ ) and θ0 = ( ) arctan (Numerically one has µ = ) In particular, we have π p q q 8.06 and π log 2 p q q 8.06 for any integers p,q with q q 0. Hence the former remarkably improves the earlier irrationality measures of π mentioned above. As another application of the integral (.3), by taking a =, a 2 = ( + 3i)/2 and a 3 = (3 + 3i)/4, we have the following

3 Rational approximations 337 Theorem.2. For any ε > 0, there exists a positive integer q 0 (ε) such that π p 3 q q ξ ε for any integers p,q with q q 0 (ε), where the exponent ξ is given by ξ = + 2log(β/54) + 6 π/ 3 log(β/3) 6 + π/ 3 with β = u + v + 3 u v, u = and v = (Numerically one has ξ = ) Concerning the number π/ 3, K. Alladi and M. L. Robinson [] obtained an irrationality measure by using the Legendre polynomials. Later this was improved to by G. V. Chudnovsky [4], to 5.56 by A. K. Dubitskas [8], to by the author [0], and to 4.97 by G. Rhin [4]. (The last measure 4.97 was announced without proof.) Therefore our Theorem.2 improves all the measures for π/ 3 mentioned above. The principal part of the integral (.3), as n tends to +, can be easily obtained by the saddle method originated in Riemann s work, which is described in detail in Dieudonné s book [7]. This may be regarded as a natural complex version of the well-known formula lim n ( 0 ϕ(t)(f(t)) n dt) /n = max 0 t f(t) for real-valued positive continuous functions ϕ(t) and f(t). Owing to this method we do not need in this paper any discussion on recurrence relations of polynomials. We note that our new exponents appearing in Theorems. and.2 contain the roots of some cubic equations with integral coefficients. We prepare four lemmas in the next section. In particular, Lemma 2.2 is an elementary arithmetical lemma concerning binomial coefficients, which plays an important role in this paper. Then Theorems. and.2 will be proved in Sections 3 and 4 respectively. 2. Preliminaries. First of all we need the following lemma concerning linear independences over Q: Lemma 2.. Let m be a fixed non-negative integer. Let γ and γ 2 be real numbers satisfying q n γ p n = ε n and q n γ 2 r n = δ n for some p n, q n, r n Z + i mz for all n. Suppose that lim n n log q n = σ, lim n n log ε n = τ, lim n n log δ n = τ

4 338 M. Hata for positive numbers σ, τ and τ with τ τ. Suppose further that there exist infinitely many n s satisfying δ n /ε n for any rational number. Then the numbers, γ and γ 2 are linearly independent over Q. More precisely, for any ε > 0, there exists a positive integer H 0 (ε) such that p + qγ + rγ 2 H σ/τ ε for any integers p, q, r with H max{ q, r } H 0 (ε). P r o o f. Obviously there exists an integer n 0 such that q n 0, ε n 0 and δ n 0 for all n n 0. Putting Λ = p + qγ + rγ 2 for arbitrarily fixed integers (p,q,r) (0,0,0), we have ( ) ( ) pn + ε n rn + δ n Λ = p + q + r q n = pq n + qp n + rr n q n q n + qε n + rδ n q n A n q n + ω n q n, say. We first show that the set Ω = {n : A n 0} is infinite. Suppose, on the contrary, that Ω is finite; that is, A n = 0 for all n n for some integer n n 0. Since q n and ω n 0 as n, we have Λ = 0. Hence ω n = 0; so, q + r δ n ε n = 0 for all n n. Since (q,r) (0,0), we have r 0; hence δ n /ε n = q/r for all n n, contrary to the hypothesis of the lemma. For any ε > 0, one can define a sufficiently small δ δ(ε) (0,τ/6) satisfying σ + δ τ 3δ < σ τ + ε 2. Then there exists an integer n(ε) n 0 such that e (σ δ)n q n e (σ+δ)n and max{ ε n, δ n } e (τ δ)n for all n n(ε). We now define and H 0 (ε) = min{n : 4N e (τ δ)n(ε) and N ετ (4e σ+τ ) 2τ+4σ } N(H) = min{n > n(ε) : 4H < e (τ δ)n } for all H H 0 (ε). Since ω n 2He (τ δ)n for all n n(ε), we have ω n < /2 for all n N(H). Therefore (2.) Λ = A n + ω n q n ω n > q n 2 q n 2 e (σ+δ)n for any n Ω with n N(H). (Note that A n if A n 0, since A n Z + i mz.)

5 Rational approximations 339 Now we distinguish two cases as follows: (a) N(H) Ω or (b) N(H) Ω. In Case (a), it follows from the definition of N(H) that 4H e (τ δ)(n(h) ) ; so, e N(H) 2 3/τ eh /(τ δ). Hence, putting n = N(H) in (2.), we have Λ 2 e (σ+δ)n(h) (H 0 (ε)) ε/2 H σ/τ ε/2 H σ/τ ε, as required. In Case (b), let M(H) be the smallest integer satisfying M(H) > N(H) and M(H) Ω. Since A M(H) = 0, we have Λ = ω M(H) 2He (σ+τ 2δ)(M(H) ) ; so, q M(H) ( ) /(σ+τ 2δ) 2H e M(H) e. Λ Hence, putting n = M(H) in (2.), we obtain thus, Λ 2 e (σ+δ)m(h) 2 e (σ+δ) ( Λ 2H This completes the proof. Λ (4e σ+τ ) 2σ/τ H (σ+δ)/(τ 3δ) ) (σ+δ)/(σ+τ 2δ) ; (H 0 (ε)) ε/2 H σ/τ ε/2 H σ/τ ε. R e m a r k 2.. By the same argument as above, one can show that if q n γ p n = ε n for some irrational number γ, lim n n log q n = σ and lim sup n n log ε n τ, then γ has an irrationality measure + σ/τ. This slightly improves Proposition 3.3 in [0], in which the author posed the unnecessary hypothesis that γ is not a Liouville number. This remark will be used in the proof of Theorem.2. We next give an elementary arithmetical lemma concerning binomial coefficients. Lemma 2.2. There exists a positive integer D n such that ( )( )( ) D n 2n 2n 2n (2.2) Z j + k + l 3n j k l

6 340 M. Hata for all integers 0 j,k,l 2n with j + k + l 3n and that (2.3) lim n n log D n = { 6 π + log 27 } κ, say (Numerically one has κ = ) P r o o f. Let v(s,p) be the exponent of any prime factor p in the resolution of each integer s 3n into its prime factors; that is, s = p v(s,p). Then clearly v(s,p) p prime [ ] log s log p [ ] log(3n). log p We next show that any prime number p > 3n belonging to the set { T n = p prime : { } n 2 < 2 }, p 3 divides all the integers ( )( )( ) 2n 2n 2n, j k l where (j,k,l) runs through the set Ξ(n,p) = {(j,k,l) : 0 j,k,l 2n and j +k +l 3n (mod p)}. ({x} denotes the fractional part of x.) To see this, it suffices to show that [ ] [ ] [ 2n j 2n j J = p p p ] + [ ] 2n p [ ] [ ] k 2n k p p [ ] [ ] [ ] 2n l 2n l + p p p for any (j,k,l) Ξ(n,p), since p > 3n. Let ω = {n/p}, θ = {j/p}, θ = {k/p} and θ = {l/p}. Then obviously J = [2ω] [2ω θ] + [2ω] [2ω θ ] + [2ω] [2ω θ ] = 3 [2ω θ] [2ω θ ] [2ω θ ], since [2ω] =. On the other hand, since 3n = j +k+l+pq for some integer q and since [3ω] =, we have {3ω} = {θ + θ + θ } = 3ω. Thus θ + θ + θ = 3ω + r for some integer r 0; hence [2ω θ] + [2ω θ ] + [2ω θ ] [6ω θ θ θ ] = [6ω 3ω + r] Therefore we have J, as required. = r + [3ω] = 2 r 2.

7 We define (n) = p prime p 3n p [ log(3n) log p Rational approximations 34 ], 2 (n) = p prime p 3n for all n. The above argument implies that the integer D n = (n) 2 (n) 3 (n) p, 3 (n) = p prime p T n satisfies (2.2) for any integers j,k,l [0,2n] with j + k + l 3n. Put b i = 3/(3i + 2), c i = 2/(2i + ) and S i (n) = log p n for i 0 for brevity. Then clearly n log 3(n) = p prime p/n (b i,c i ] S i (n) and it follows from the prime number theorem that S i (n) c i b i as n for each i 0. Thus lim n n log 3(n) = i=0 i=0 lim S i(n) = n (c i b i ) i=0 = Γ (2/3) Γ(2/3) Γ (/2) Γ(/2) = 2 { π 3 } 27 log, 6 where Γ(z) is the gamma function. (The interchange of the operations of limit and summation can be easily justified by using the estimate S i (n) const c L i L for any L, where const depends neither on L nor on n.) Since it can be seen that log (n) = O( n) and log 2 (n) 3n as n tends to +, we thus obtain the asymptotic behaviour (2.3). This completes the proof. We now consider the function F(z) F(a,a 2,a 3 ;z) defined in (.4) for non-zero distinct complex numbers a, a 2 and a 3. It is easily seen that F (z) has three zero points ζ, ζ 2, ζ 3 in the region C {a,a 2,a 3 }, which satisfy the cubic equation (2.4) z 3 + s 2 z 2 + s z + s 0 = 0 p

8 342 M. Hata with s 0 = a a 2 a 3, s = (a a 2 +a 2 a 3 +a 3 a )/3 and s 2 = (a +a 2 +a 3 )/3. Such points ζ j are called saddles of the surface Σ of F(z). Using the equalities { } F(ζ j ) = 4 (s 2 2 3s )ζ j + s2 3s 0 s 2 7s s 2 s 0 ζ j for j 3, one can easily calculate every elementary symmetric function on F(ζ ), F(ζ 2 ) and F(ζ 3 ). We thus have Lemma 2.3. Put t 0 = s 0, t = s s 2 and t 2 = s 3 2+s 3 /s 0 for brevity. Then we have the identity where 3 (z F(ζ j )) = z 3 + 4σ 2 z 2 + 6σ z + 64σ 0, j= σ 0 = t 3 0 2t 2 0t 8t 2 0t t 0 t t 0 t t 2 + 6t 0 t t t 2 t 2 + 9t 4 /t 0, σ = 3t t 0t + 20t 0 t t 2 + 2t t 2 + t 3 /t 0, σ 2 = 3t 0 + 5t + t 2. Finally, we recall the following result from Dieudonné s book [7]: Lemma 2.4. Let Γ be a smooth oriented path with a finite length, which departs from a j and arrives at a k through a saddle ζ l for some j,k,l 3. Put l z = {z + t : t 0}, the half-straight line parallel to the real axis. Suppose that the saddle ζ l is a simple root of (2.4) and suppose further that Γ {a j,a k } is contained in the simply connected region C 0 C (l 0 l a l a2 l a3 ) and that Γ {a j,a k,ζ l } is contained in the open set {z C 0 : F(z) < F(ζ l ) }. Then the principal part of the integral as n tends to +, is given by in particular, we have I n = Γ (F(z)) n dz z, I n d 0 (F(ζ l )) n n with d 0 = lim n n log I n = log F(ζ l ). 2π F(ζ l) F (ζ l ) ζ l ; Indeed, the above lemma is an immediate consequence of the result in

9 Rational approximations 343 [7; Chapter IX], since the integral I n can be expressed as I n = ( ) n Γ g(z)e nh(z) dz, where g(z) = /z and h(z) = 2 3 j= Log(a j z) 3Log( z) are analytic functions in C Proof of Theorem.. Let Γ z,w be a smooth oriented path departing from z, arriving at w, and contained in C {0}. We then consider the integral where I n (Γ z,w ) = Γ z,w 2n 2n j=0 k=0 l=0 (F(a,a 2,a 3 ;z)) n dz z 2n B j,k,l ( 2n j k l ) Γ z,w B j,k,l = ( ) j+k+l a 2n j a 2n k 2 a 2n l 3. Therefore we have the identity (3.) I n (Γ z,w ) = j+k+l 3n + j+k+l=3n u n (z,w) + v n B j,k,l j + k + l 3n B j,k,l ( 2n j Γ z,w dz z, ( 2n j k say. k l l ) Γ z,w z j+k+l 3n dz, ) (w j+k+l 3n z j+k+l 3n ) For the proof of Theorem. we choose a =, a 2 = 2 and a 3 = + i. Then it follows from Lemma 2.3 that w k 9 4 (i )F(ζ k), k 3, satisfy the cubic equation dz z 9w w w 36 = 0. This algebraic equation has three positive roots; indeed, w k = cos (θ 0 + 2kπ ) for k 3 where θ 0 = 3 arctan ( ). We thus have F(ζk ) = 2 9 2wk. Let Γ,2 and Γ,+i be the paths illustrated in Figure through

10 344 M. Hata the saddles ζ and ζ 2 respectively. Then clearly dz z = log 2 and dz z = 2 log 2 + π 4 i. Γ,2 Γ,+i Fig. Since and ( ) j+k+l B j,k,l 2 j+k+l 3n = 2 j+l n ( + i) 2n l = 2 j i n ( i) l ( ) j+k+l B j,k,l (+i) j+k+l 3n = 2 2n k (+i) j+k n = (+i) j+n ( i) 2n k, it follows from Lemma 2.2 that both u n (,2) and u n (, + i) belong to the set (Z + iz)/d n. Hence, putting p n = 2iD n {u n (,2) 2u n (, + i)}, q n = D n v n, r n = D n u n (,2), we obtain and q n π p n = 2iD n {I n (Γ,2 ) 2I n (Γ,+i )} ε n q n log 2 r n = D n I n (Γ,2 ) δ n, where p n,q n,r n Z + iz. Then it follows from Lemmas 2.2 and 2.4 that say, τ = lim n n log ε n = κ log F(ζ 2 )

11 Rational approximations 345 and τ = lim n n log δ n = κ log F(ζ ), since F(ζ ) < F(ζ 2 ). Moreover, the hypothesis on the sequence {δ n /ε n } in Lemma 2. is clearly satisfied since τ > τ. Fig. 2 On the other hand, v n = ( 2n B j,k,l j j+k+l=3n k l ) = 2πi C (F(z)) n dz z, where C = Γ,+i Γ +i, is a closed oriented curve enclosing the origin and Γ +i, is the path illustrated in Figure 2 through the saddle ζ 3. Hence σ = lim n n log q n = κ + log F(ζ 3 ), since F(ζ 3 ) > F(ζ 2 ). Therefore, by Lemma 2., the numbers,log 2 and π have a linear independence measure This completes the proof. µ = σ τ = κ + log F(ζ 3) κ + log F(ζ 2 ). 4. Proof of Theorem.2. We take a =, a 2 = λ, a 3 = λ = (+λ)/2 with λ = e πi/3. Then it follows from Lemma 2.3 that w k = 3 3iF(ζ k ),

12 346 M. Hata k 3, satisfy the cubic equation 8w w 2 + 8w = 0. Indeed, this algebraic equation has one positive root w 3 = 54 ( u + v + 3 u v) β 54, say, with u = and v = , and two complex roots w,w 2 = w. We thus have 3 3β F(ζ 3 ) = 9 w 3 = and F(ζ ) = F(ζ 2 ) = β since w w 2 w 3 = /8. Let Γ,λ be the path illustrated in Figure 3 through the saddles ζ and ζ 2. Then clearly and Γ,λ dz z = Log λ = π 3 i ( ) j+k+l B j,k,l = λ 2n k ( + λ) 2n l 2 l 2n Z + i 3Z 2 2n+. Fig. 3 Hence, putting q n = 2 2n+ D n v n and p n = 2 2n+ 3iD n u n (,λ),

13 Rational approximations 347 we obtain, from (3.), q n π 3 p n = 2 2n+ 3iD n I n (Γ,λ ) ε n, say, where p n,q n Z + i 3Z. Then it follows from Lemmas 2.2 and 2.4 that lim sup n and that since n log ε n log 4 + κ + lim sup n n log( I n(γ,λ ) + I n (Γ λ,λ) ) log 4 + κ + log F(ζ ) τ, say, σ = lim n n log q n = log 4 + κ + log F(ζ 3 ) v n = 2πi C (F(z)) n dz z where C = Γ,λ Γ λ, is a closed curve enclosing the origin and Γ λ, is the path illustrated in Figure 4 through the saddle ζ 3. Therefore, by Remark 2., the number π/ 3 has an irrationality measure ξ = + σ τ = log 4 + κ + log F(ζ 3) log 4 + κ + log F(ζ ). This completes the proof of Theorem.2. Fig. 4

14 348 M. Hata 5. Remarks. Using Lemma 2. one can also obtain a linear independence measure for the numbers,π/ 3 and log(3/4) from the integral considered in Section 4. However, it can be seen that the case in which a = 3/2, a 2 = 2 and a 3 = + λ with λ = e πi/3 gives a better linear independence measure µ for such numbers. Indeed, we have µ = log 8 + κ + log x 3 log 8 + κ + log x 2 = , where w k = (/ 3)e πi/6 x k, k 3, are three positive roots of the cubic equation 8748w w w = 0 satisfying w < w 2 < w 3. The hypothesis on {δ n /ε n } in Lemma 2. is clearly satisfied since τ > τ. (We must multiply both sides of (3.) by the factor 2 3n+ D n.) In particular, π p 3log(3/4) q q for any integer p and for sufficiently large integer q. Similarly one can obtain a linear independence measure µ for the numbers, π/ 3 and log 3 by taking a =, a 2 = λ and a 3 = + λ. In fact, µ 2 = log 3 + κ + log y 3 2 log 3 + κ + log y = , where w k = 4 3 3iyk, k 3, are three roots of the cubic equation 9w 3 87w 2 6w = 0 satisfying w 3 > 0 and w = w 2. (We must multiply both sides of (3.) by the factor 2(+λ) n D n.) In this case we have y = y 2 (that is, τ = τ); so it follows from Lemma 2.4 that δ n = d e inν + o() ε n as n tends to +, where d 0 is some complex constant and ν = arg w arg w 2 satisfies 0 < ν < π. Hence the sequence {δ n /ε n } satisfies the hypothesis of Lemma 2., as required. In particular, π p 3 log 3 q q for any integer p and for sufficiently large q. References [] K. Alladi and M. L. Robinson, Legendre polynomials and irrationality, J. Reine Angew. Math. 38 (980), [2] F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. (979),

15 Rational approximations 349 [3] G. V. Chudnovsky, Hermite Padé approximations to exponential functions and elementary estimates of the measure of irrationality of π, in: Lecture Notes in Math. 925, Springer, Berlin 982, [4], Recurrences, Padé approximations and their applications, in: Lecture Notes in Pure and Appl. Math. 92, Dekker, New York 984, [5] D. V. Chudnovsky and G. V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, in: Lecture Notes in Math. 052, Springer, Berlin 984, [6],, Transcendental methods and theta-functions, in: Proc. Sympos. Pure Math. 49 (2), Amer. Math. Soc., 989, [7] J. D i e u d o n n é, Calcul infinitésimal, Hermann, Paris 968. [8] A. K. Dubitskas, Approximation of π/ 3 by rational fractions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 987 (6), (in Russian). [9] R. Dvornicich and C. Viola, Some remarks on Beukers integrals, in: Colloq. Math. Soc. János Bolyai 5, Budapest 987, [0] M. H a t a, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (990), [], A lower bound for rational approximations to π, J. Number Theory, to appear. [2] K. M a h l e r, On the approximation of π, Nederl. Akad. Wetensch. Proc. Ser. A 56 (953), [3] M. M i g n o t t e, Approximations rationnelles de π et quelques autres nombres, Bull. Soc. Math. France Mém. 37 (974), [4] G. R h i n, Approximants de Padé et mesures effectives d irrationalité, in: Progr. in Math. 7, Birkhäuser, 987, [5] E. A. Rukhadze, A lower bound for the approximation of ln 2 by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 987 (6), (in Russian). INSTITUTE OF MATHEMATICS YOSHIDA COLLEGE KYOTO UNIVERSITY KYOTO 606, JAPAN Received on and in revised form on (2239)