aa r X i v : . [ m a t h . G M ] M a r Irrationality Exponents For Even Zeta Constants
N. A. Carella
Abstract : Let k ≥ (cid:12)(cid:12) p/q − π k (cid:12)(cid:12) > /q µ ( π k ) of the irrational number π k are bounded away from zero. A general result for the irrationalityexponent µ ( π k ) will be proved here. The specific results and numerical data for a few cases k = 2and k = 3 are also presented and explained. The even parameters 2 k correspond to the even zetaconstants ζ (2 k ). Contents π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Exponent For The Number π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Exponent For The General Case π k . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 (cid:12)(cid:12) / sin π k +1 z (cid:12)(cid:12)
44 Upper Bound For (cid:12)(cid:12) / sin π z (cid:12)(cid:12)
65 Upper Bound For / (cid:12)(cid:12) sin π z (cid:12)(cid:12)
76 The Exponent Result For π µ ( π ) . . . . . . . . . . . . . . . . . . . . . . . 9 π µ ( π ) . . . . . . . . . . . . . . . . . . . . . . . 10 ζ (3) µ ( ζ (3)) . . . . . . . . . . . . . . . . . . . . . . 11 π k List of Tables µ ( π ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Numerical Data For The Exponent µ ( π ) . . . . . . . . . . . . . . . . . . . . . . . 103 Numerical Data For The Exponent µ ( π ) . . . . . . . . . . . . . . . . . . . . . . . 114 Historical Data For µ ( ζ (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Numerical Data For The Exponent µ ( ζ (3)) . . . . . . . . . . . . . . . . . . . . . . 12 March 4, 2020
AMS MSC : Primary 11J82, Secondary 11J72; 11Y60.
Keywords : Irrational number; Irrationality exponent; Pi. rrationality Exponents For Even Zeta Constants Let k ≥ (cid:12)(cid:12) p/q − π k (cid:12)(cid:12) > /q µ ( π k ) of theirrational number π k are bounded away from zero. The earliest result | p/q − π | > /q for theirrationality exponent µ ( π ) was proved by Mahler in 1953, and more recently it was reduced to | p/q − π | > /q . by Salikhov in 2008. The earliest result for next number (cid:12)(cid:12) p/q − π (cid:12)(cid:12) > /q . was proved by Apery in 1979, and more recently it was reduced to (cid:12)(cid:12) p/q − π (cid:12)(cid:12) > /q . by Rhinand Viola in 1996. Therer is no literature for k ≥
3. This note introduces elementary techniquesto determine the irrationality exponent µ ( π k ) of the irrational number π k . It is shown that theDiophantine inequality (cid:12)(cid:12) p/q − π k (cid:12)(cid:12) > /q ε , where ε > k ≥ π Let { p n /q n : n ≥ } be the sequence of convergents of the irrational number π . The sequence ofrational approximations { (cid:12)(cid:12) p n /q n − π (cid:12)(cid:12) : n ≥ } are bounded away from zero. For instance, the 5thand 6th convergents are(i) (cid:12)(cid:12)(cid:12)(cid:12) − π (cid:12)(cid:12)(cid:12)(cid:12) ≥ . , (ii) (cid:12)(cid:12)(cid:12)(cid:12) − π (cid:12)(cid:12)(cid:12)(cid:12) ≥ . ,respectively, additional data are compiled in Table 2. But, it is difficult to prove a lower bound.The earliest result (cid:12)(cid:12) p/q − π (cid:12)(cid:12) ≥ /q . was proved by Apery in [1], and more recently it wasimproved to (cid:12)(cid:12) p/q − π (cid:12)(cid:12) ≥ /q . by Rhin and Viola in [22]. Basic and elementary ideas are usedhere to improve it to the followings estimate. Theorem 1.1.
For all large rational approximations p/q → π , the Diophantine inequality (cid:12)(cid:12)(cid:12)(cid:12) π − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ q ε , (1) where ε > is a small number, is true. The proof appears in Section 6.Table 1: Historical Data For µ ( π )Irrationality Measure Upper Bound Reference Year µ ( π ) ≤ . µ ( π ) ≤ . µ ( π ) ≤ . π Let { p n /q n : n ≥ } be the sequence of convergents of the irrational number π . The sequence ofrational approximations { (cid:12)(cid:12) p n /q n − π (cid:12)(cid:12) : n ≥ } are bounded away from zero. For instance, the 5thand 6th convergents are(i) (cid:12)(cid:12)(cid:12)(cid:12) − π (cid:12)(cid:12)(cid:12)(cid:12) ≥ . , (ii) (cid:12)(cid:12)(cid:12)(cid:12) − π (cid:12)(cid:12)(cid:12)(cid:12) ≥ . ,respectively, additional data are compiled Table 3. But, it is difficult to prove a lower bound.The literature does not have any estimate nor numerical data on the irrationality exponent of thisnumber. Basic and elementary ideas are used here to prove the followings estimate. rrationality Exponents For Even Zeta Constants Theorem 1.2.
For all large rational approximations p/q → π , the Diophantine inequality (cid:12)(cid:12)(cid:12)(cid:12) π − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ q ε , (2) where ε > is a small number, is true. The proof appears in Section 7. π k Theorem 1.3.
Let k ≥ be a small fixed integer. For all large rational approximations p/q → π k ,the Diophantine inequality (cid:12)(cid:12)(cid:12)(cid:12) π k − pq (cid:12)(cid:12)(cid:12)(cid:12) ≫ q ε , (3) where ε > is a small number, is true. The proof appears in Section 9.
The harmonic summation kernels naturally arise in the partial sums of Fourier series and in thestudies of convergences of continuous functions.
Definition 2.1.
The Dirichlet kernel is defined by D x ( z ) = X − x ≤ n ≤ x e i nz = sin((2 x + 1) z )sin ( z ) , (4) where x ∈ N is an integer and z ∈ R − π Z is a real number. Definition 2.2.
The Fejer kernel is defined by F x ( z ) = X ≤ n ≤ x, X − n ≤ k ≤ n e i kz = 12 sin(( x + 1) z ) sin ( z ) , (5) where x ∈ N is an integer and z ∈ R − π Z is a real number. These formulas are well known, see [15] and similar references. For z = kπ , the harmonic summa-tion kernels have the upper bounds |K x ( z ) | = |D x ( z ) | ≪ | x | , and |K x ( z ) | = |F x ( z ) | ≪ | x | .The Dirichlet kernel in Definition 2.1 is a well defined continued function of two variables x, z ∈ R .Hence, for fixed z , it has an analytic continuation to all the real numbers x ∈ R .An important property is the that a proper choice of the parameter x ≥ / sin z to K x ( z ), and the term 1 / sin(2 x + 1) z remainsbounded. This principle will be applied to certain lacunary sequences { q n : n ≥ } , which maxi-mize the reciprocal sine function 1 / sin z , to obtain an effective upper bound of the function 1 / sin z . Lemma 2.1.
Let k ≥ be a small fixed integer, and let { p n /q n : n ≥ } be the sequence ofconvergents of the real number π k , and = z ∈ Z . Then | sin( π k +1 z ) | ≪ | sin ( π k +1 q n ) | . (6) rrationality Exponents For Even Zeta Constants Proof.
By the best approximation principle, see Lemma 10.6, (cid:12)(cid:12) m − π k z (cid:12)(cid:12) ≥ (cid:12)(cid:12) p n − π k q n (cid:12)(cid:12) (7)for any integer z ≤ q n . Hence, 1 | sin ( π k +1 z ) | = 1 | sin ( πm − π k +1 z ) | (8) ≤ | sin ( πp n − π k +1 q n ) | = 1 | sin ( π k +1 q n ) | , as n → ∞ . (cid:4) (cid:12)(cid:12) / sin π k +1 z (cid:12)(cid:12) As shown in Lemma 2.1, to estimate the upper bound of the function 1 / | sin π k +1 z | over the realnumbers z ∈ R , it is sufficient to fix z = q n , and select a real number x ∈ R such that q n ≍ x .This idea is demonstrated below for small integer parameter k ≥ Lemma 3.1.
Let k ≥ be a small fixed integer, let { p n /q n : n ≥ } be the sequence of convergentsof the real number π k , and define the associated sequence x n = (cid:18) v + 12 v (cid:19) q n π k , (9) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (i) sin (cid:0) x n − /
2) + 1) π k +1 q n (cid:1) = ± . (ii) sin (cid:0) x n + 1 /
2) + 1) π k +1 q n (cid:1) = ± cos 2 π k +1 q n . (iii) (cid:12)(cid:12) sin (cid:0) x n + 1 / π k +1 q n (cid:1)(cid:12)(cid:12) ≥ − π q n , as n → ∞ .Proof. Observe that the value x n in (9) yieldssin(2 π k +1 q n x n ) = sin (cid:18) π k +1 q n (cid:18) v + 12 v (cid:19) q n π k (cid:19) = sin (cid:16) π · w n (cid:17) = ± , (10)and cos (cid:0) π k +1 q n x n (cid:1) = cos (cid:18) π k +1 q n (cid:18) v + 12 v (cid:19) q n π k (cid:19) = cos (cid:16) π · w n (cid:17) = 0 , (11)where w n = (cid:18) v + 12 v (cid:19) q n (12)is an odd integer. (i) Routine calculations yield this:sin((2( x n − /
2) + 1) π k +1 q n ) = sin (cid:0) π k +1 q n x n (cid:1) (13)= sin (cid:18) π k +1 q n (cid:18) v + 12 v (cid:19) q n π k (cid:19) = sin (cid:16) π · w n (cid:17) = ± , rrationality Exponents For Even Zeta Constants (cid:0) (2( x n + 1 /
2) + 1) π k +1 q n (cid:1) = sin(2 π k +1 q n x n + 2 π k +1 q n ) (14)= sin(2 π k +1 q n x n ) cos(2 π k +1 q n )+ cos(2 π k +1 q n x n ) sin(2 π k +1 q n ) . Substituting (10) and (11) into (14) returnsin (cid:0) x n + 1 /
2) + 1) π k +1 q n (cid:1) = ± cos (cid:0) π k +1 q n (cid:1) . (15)(iii) This follows from the previous result: (cid:12)(cid:12) sin (cid:0) x n + 1 /
2) + 1) π k +1 q n (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ± cos (cid:0) π k +1 q n (cid:1)(cid:12)(cid:12) (16)= (cid:12)(cid:12) ± cos (cid:0) πp n − π k +1 q n (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ± cos (cid:0) π (cid:0) p n − π k q n (cid:1)(cid:1)(cid:12)(cid:12) ≍ , since the sequence of convergents satisfies (cid:12)(cid:12) p n − π k q n (cid:12)(cid:12) ≤ /q n as n → ∞ . (cid:4) Lemma 3.2.
Let k ≥ be a small fixed integer, let { p n /q n : n ≥ } be the sequence of convergentsof the real number π k , and define the associated sequence x n = (cid:18) v + 12 v (cid:19) q n π k , (17) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (cid:12)(cid:12) sin (cid:0) x ∗ + 1) π k +1 q n (cid:1)(cid:12)(cid:12) ≍ , (18) where x ∗ ∈ [ x n − / , x n + 1 / is an integer.Proof. Consider the continuous function f ( x ) = (cid:12)(cid:12) sin (cid:0) x + 1) π k +1 q n (cid:1)(cid:12)(cid:12) over the interval [ x n − / , x n + 1 / x = x n − / ∈ R : (cid:12)(cid:12) sin (cid:0) (2 x + 1) π k +1 z (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) (2( x n − /
2) + 1) π k +1 q n (cid:1)(cid:12)(cid:12) (19)= 1 , and it has a local minimal at x = x n + 1 / ∈ R : (cid:12)(cid:12) sin (cid:0) (2 x + 1) π k +1 z (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) (2( x n + 1 /
2) + 1) π k +1 q n (cid:1)(cid:12)(cid:12) (20) ≥ − π q n . Since f ( x ) is continuous over the interval [ x n − / , x n + 1 / − π q n ≤ (cid:12)(cid:12) sin (cid:0) (2 x ∗ + 1) π k +1 z (cid:1)(cid:12)(cid:12) ≤ x ∗ ∈ [ x n − / , x n + 1 / (cid:4) Theorem 3.1. If k ≥ is a small fixed integer, and z ∈ N is a large integer, then, (cid:12)(cid:12)(cid:12)(cid:12) π k +1 z (cid:12)(cid:12)(cid:12)(cid:12) ≪ | z | . (21) Proof.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π k . Since thedenominators sequence { q n : n ≥ } maximize the reciprocal sine function 1 / sin π k +1 z , see Lemma2.1, it is sufficient to prove it for z = q n . Define the associated sequence x n = (cid:18) v + 12 v (cid:19) q n π k , (22) rrationality Exponents For Even Zeta Constants v = v ( q n ) = max { v : 2 v | q n } is the 2-adic valuation, and n ≥
1. Let f ( x ) = (cid:12)(cid:12) sin (cid:0) (2 x + 1) π k +1 z (cid:1)(cid:12)(cid:12) , and let z = q n . The function f ( x ) is bounded over the interval [ x n − / , x n + 1 / x ∗ ∈ [ x n − / , x n + 1 / z = q n ,and applying Lemma 3.1 return (cid:12)(cid:12) sin (cid:0) (2 x + 1) π k +1 z (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) (2 x ∗ + 1) π k +1 q n (cid:1)(cid:12)(cid:12) (23) ≍ . Rewrite the reciprocal sine function in terms of the harmonic kernel in Definition 2.1, and spliceall these information together, to obtain (cid:12)(cid:12)(cid:12)(cid:12) π k +1 z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D x ( π k +1 z )sin((2 x + 1) π k +1 z ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ |D x ∗ | (cid:12)(cid:12)(cid:12)(cid:12) x ∗ + 1) π k +1 q n ) (cid:12)(cid:12)(cid:12)(cid:12) (24) ≪ | x ∗ | · ≪ | z | since | z |≍ x ∗ ≍ p n ≍ q n , and the trivial estimate |D x ( z ) | ≪ | x | . (cid:4) (cid:12)(cid:12) / sin π z (cid:12)(cid:12) As shown in Lemma 2.1, to estimate the upper bound of the function 1 / | sin π z | over the realnumbers z ∈ R , it is sufficient to fix z = q n , and select a real number x ∈ R such that q n ≍ x .This idea is demonstrated below. Lemma 4.1.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π , and definethe associated sequence x n = (cid:18) v + 12 v (cid:19) q n π , (25) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (i) sin (cid:0) x n − /
2) + 1) π q n (cid:1) = ± . (ii) sin (cid:0) x n + 1 /
2) + 1) π q n (cid:1) = ± cos 2 π q n . (iii) (cid:12)(cid:12) sin (cid:0) x n + 1 / π q n (cid:1)(cid:12)(cid:12) ≥ − π q n , as x → ∞ .Proof. Same as Lemma 3.1. (cid:4)
Lemma 4.2.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π , and definethe associated sequence x n = (cid:18) v + 12 v (cid:19) q n π k , (26) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (cid:12)(cid:12) sin (cid:0) x ∗ + 1) π q n (cid:1)(cid:12)(cid:12) ≍ , (27) where x ∗ ∈ [ x n − / , x n + 1 / is an integer.Proof. Same as Lemma 3.2. (cid:4)
Theorem 4.1.
Let z ∈ N be a large integer. Then, (cid:12)(cid:12)(cid:12)(cid:12) π z (cid:12)(cid:12)(cid:12)(cid:12) ≪ | z | . (28) rrationality Exponents For Even Zeta Constants Proof.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π . Since thedenominators sequence { q n : n ≥ } maximize the reciprocal sine function 1 / sin π z , it is sufficientto prove it for z = q n . Define the associated sequence x n = (cid:18) v + 12 v (cid:19) q n π , (29)where v = v ( q n ) = max { v : 2 v | q n } is the 2-adic valuation, and n ≥
1. Replacing the integerparameters x ∗ ∈ [ x n − / , x n + 1 / z = q n , and applying Lemma 4.2 return (cid:12)(cid:12) sin (cid:0) (2 x + 1) π z (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) sin (cid:0) (2 x ∗ + 1) π q n (cid:1)(cid:12)(cid:12) (30) ≍ , since the sequence of convergents satisfies (cid:12)(cid:12) p n − π q n (cid:12)(cid:12) → n → ∞ . Rewrite the reciprocal sinefunction in terms of the harmonic kernel in Definition 2.1, and splice all these information together,to obtain (cid:12)(cid:12)(cid:12)(cid:12) π z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D x ( π z )sin((2 x + 1) π z ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ |D x ∗ | (cid:12)(cid:12)(cid:12)(cid:12) x ∗ + 1) π q n ) (cid:12)(cid:12)(cid:12)(cid:12) (31) ≪ | x ∗ | · ≪ | z | since | z |≍ x ∗ ≍ p n ≍ q n , and the trivial estimate |D x ( z ) | ≪ | x | . (cid:4) / (cid:12)(cid:12) sin π z (cid:12)(cid:12) As shown in Lemma 2.1, to estimate the upper bound of the function 1 / | sin π z | over the realnumbers z ∈ R , it is sufficient to fix z = q n , and select a real number x ∈ R such that q n ≍ x .This idea is demonstrated below. Lemma 5.1.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π , and definethe associated sequence x n = (cid:18) v + 12 v (cid:19) q n π , (32) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (i) sin (cid:0) x n − /
2) + 1) π q n (cid:1) = ± . (ii) sin (cid:0) x n + 1 /
2) + 1) π q n (cid:1) = ± cos 2 π q n . (iii) (cid:12)(cid:12) sin (cid:0) x n + 1 / π q n (cid:1)(cid:12)(cid:12) ≥ − π q n , as x → ∞ .Proof. Same as Lemma 3.1. (cid:4)
Lemma 5.2.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π , and definethe associated sequence x n = (cid:18) v + 12 v (cid:19) q n π , (33) where v = v ( q n ) = max { v : 2 v | q n } is the -adic valuation, and n ≥ . Then (cid:12)(cid:12) sin (cid:0) x ∗ + 1) π q n (cid:1)(cid:12)(cid:12) ≍ , (34) where x ∗ ∈ [ x n − / , x n + 1 / is an integer.Proof. Same as Lemma 3.2. (cid:4) rrationality Exponents For Even Zeta Constants Theorem 5.1.
Let z ∈ N be a large integer. Then, (cid:12)(cid:12)(cid:12)(cid:12) π z (cid:12)(cid:12)(cid:12)(cid:12) ≪ | z | . (35) Proof.
Let { p n /q n : n ≥ } be the sequence of convergents of the real number π . Since thedenominators sequence { q n : n ≥ } maximize the reciprocal sine function 1 / sin π z , it is sufficientto prove it for z = q n . Define the associated sequence x n = (cid:18) v + 12 v (cid:19) q n π , (36)where v = v ( q n ) = max { v : 2 v | q n } is the 2-adic valuation, and n ≥
1. Replacing the integerparameters x ∗ ∈ [ x n − / , x n + 1 / z = q n , and applying Lemma 5.2 return | sin ((2 x + 1) z ) | = (cid:12)(cid:12) sin (cid:0) (2 x ∗ + 1) π q n (cid:1)(cid:12)(cid:12) (37) ≍ , since the sequence of convergents satisfies (cid:12)(cid:12) p n − π q n (cid:12)(cid:12) → n → ∞ . Rewrite the reciprocal sinefunction in terms of the harmonic kernel in Definition 2.1, and splice all these information together,to obtain (cid:12)(cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D x ( z )sin((2 x + 1) z ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ |D x ∗ | (cid:12)(cid:12)(cid:12)(cid:12) x ∗ + 1) π q n ) (cid:12)(cid:12)(cid:12)(cid:12) (38) ≪ | x ∗ | · ≪ | z | since | z |≍ x ∗ ≍ p n ≍ q n , and the trivial estimate |D x ( z ) | ≪ | x | . (cid:4) π ζ (2) = π / A + B ζ (2) = Z Z x h (1 − x ) i y j (1 − y ) k (1 − xy ) i + j − l dx dy − xy , (39)where A , B ∈ Z are integers. The analysis appears in [23], and an expanded version of thetheory of cellular integrals is presented in [2, Section 5.3]. These techniques also rely on rationalfunctions approximations of π and the prime number theorem. Some relevant references are [23],[12], [11], [25], and [6] for an introduction to the rational approximations of π and the various proofs.Since ζ (2) and π have the same irrationality exponent, the analysis is done for the simpler number.The proof within is based on an effective upper bound of the reciprocal sine function over thesequence { q n : n ≥ } as derived in Section 4. Proof. (Theorem 1.1) Let ε > { p n /q n : n ≥ } be thesequence of convergents of the irrational number π . By Theorem 4.1, the reciprocal sine functionhas the upper bound (cid:12)(cid:12)(cid:12)(cid:12) π q n ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ q εn . (40) rrationality Exponents For Even Zeta Constants (cid:0) π q n (cid:1) = sin (cid:0) αp − π q n (cid:1) if and only if αp = πp n , where p n is an integer. Theseinformation lead to the following relation.1 q εn ≪ (cid:12)(cid:12) sin (cid:0) π q n (cid:1)(cid:12)(cid:12) (41) ≪ (cid:12)(cid:12) sin (cid:0) π q n − πp n (cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) sin (cid:0) π (cid:0) π q n − p n (cid:1)(cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) π q n − p n (cid:12)(cid:12) for all sufficiently large p n /q n . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) π − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≫ q ε (42)= 1 q µ ( π )+ ε . Clearly, this implies that the irrationality measure of the real number π is µ ( π ) = 2, see Definition10.1. Quod erat demontrandum. (cid:4) µ ( π ) The continued fraction is π = [9; 1 , , , , , , , , , , , , , , , , , , , , , , , , , , . . . ] . (43)The sequence of convergents { p n /q n : n ≥ } is computed via the recursive formula provided inLemma 10.1. The approximation µ n ( π ) of the exponent in the inequality (cid:12)(cid:12)(cid:12)(cid:12) π − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≥ q µ n ( π ) (44)are tabulated in Table 2 for the early stage of the sequence of convergents p n /q n −→ π . π µ ( π ) ≥ π . Proof. (Theorem 1.2) Let ε > { p n /q n : n ≥ } be thesequence of convergents of the irrational number π . By Theorem 5.1, the reciprocal sine functionhas the upper bound (cid:12)(cid:12)(cid:12)(cid:12) π q n ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ q εn . (45)Moreover, sin (cid:0) π q n (cid:1) = sin (cid:0) αp − π q n (cid:1) if and only if αp = πp n , where p n is an integer. Theseinformation lead to the following relation.1 q εn ≪ (cid:12)(cid:12) sin (cid:0) π q n (cid:1)(cid:12)(cid:12) (46) ≪ (cid:12)(cid:12) sin (cid:0) π q n − πp n (cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) sin (cid:0) π (cid:12)(cid:12) π q n − p n (cid:12)(cid:12)(cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) π q n − p n (cid:12)(cid:12) for all sufficiently large p n /q n . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) π − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≫ q ε (47)= 1 q µ ( π )+ ε . rrationality Exponents For Even Zeta Constants µ ( π ) n p n q n µ n ( π )1 9 12 10 13 69 7 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . π is µ ( π ) = 2, see Definition10.1. Quod erat demontrandum. (cid:4) µ ( π ) The continued fraction of the second odd power of π is π = [31; 159 , , , , , , , , , , , , , , , , , , , , ... . . . ] . (48)The sequence of convergents { p n /q n : n ≥ } is computed via the recursive formula provided inLemma 10.1. The approximation µ n ( π ) of the exponent in the inequality (cid:12)(cid:12)(cid:12)(cid:12) π − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≥ q µ n ( π ) (49)are tabulated in Table 3 for the early stage of the sequence of convergents p n /q n −→ π . ζ (3) The last estimate for irrationality exponent of the odd zeta constant ζ (3) was derived from thealgebraic properties of the cellular integral A + B ζ (3) = Z Z Z x h (1 − x ) l y s z j (1 − z ) q (1 − (1 − xy ) z ) q + h − r dx dy dz (1 − (1 − xy ) z , (50) rrationality Exponents For Even Zeta Constants µ ( π ) n p n q n µ n ( π )1 31 12 4930 159 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A , B ∈ Z are integers. The analysis appears in [22], and an expanded version of the theoryof cellular integrals is presented in [2, Section 5.3].Table 4: Historical Data For µ ( ζ (3)Irrationality Measure Upper Bound Reference Year µ ( ζ (3) ≤ . µ ( ζ (3) ≤ . µ ( ζ (3) ≤ . ζ (3) and π , but is not clear if µ ( ζ (3)) = 2. In [9],it was proved that ζ (3) = απ , where α ∈ R is irrational. The numerical data in Table 5 suggeststhe followings. Conjecture 8.1.
The irrationanlity exponent of the first odd zeta constant is µ ( ζ (3)) = µ ( απ ) =2 , where α = 0 is a unique irrational number. µ ( ζ (3)) The continued fraction of the first odd zeta constant is ζ (3) = [1 , , , , , , , , , , , , , , , , , , , , , , , , , . . . ] , (51) rrationality Exponents For Even Zeta Constants { p n /q n : n ≥ } is computed via therecursive formula provided in Lemma 10.1. The approximation µ n ( ζ (3)) of the exponent in theinequality (cid:12)(cid:12)(cid:12)(cid:12) π − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≥ q µ n ( π ) (52)are tabulated in Table 5 for the early stage of the sequence of convergents p n /q n −→ π .Table 5: Numerical Data For The Exponent µ ( ζ (3)) n p n q n µ n ( ζ (3))1 1 12 5 4 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . π k The method used to prove the irrationality measure µ ( π k ) of the number π k is not based on ratio-nal functions approximations of π k and the prime number theorem. Some relevant references are[23], [12], [17], [18], [8], [11], [25], and [6] for an introduction to the rational approximations of π and the various proofs.The proof is based on an effective upper bound of the reciprocal sine function over the sequenceof { q n : n ≥ } derived in Section 3. Proof. (Theorem 1.3) Let ε > { p n /q n : n ≥ } be thesequence of convergents of the irrational number π k , with k ≥
1. By Theorem 3.1, the reciprocal rrationality Exponents For Even Zeta Constants (cid:12)(cid:12)(cid:12)(cid:12) π k +1 q n ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ q εn . (53)Moreover, the relation sin (cid:0) π k +1 q n (cid:1) = sin (cid:0) αp − π k +1 q n (cid:1) is true if and only if αp = πp n , where p n is an integer. These information lead to the following inequalities1 q εn ≪ (cid:12)(cid:12) sin (cid:0) π k +1 q n (cid:1)(cid:12)(cid:12) (54) ≪ (cid:12)(cid:12) sin (cid:0) π k +1 q n − πp n (cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) sin (cid:0) π (cid:0) π k q n − p n (cid:1)(cid:1)(cid:12)(cid:12) ≪ (cid:12)(cid:12) π k q n − p n (cid:12)(cid:12) for all sufficiently large p n /q n . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) π k − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≫ q ε (55)= 1 q µ ( π k )+ ε . Clearly, this implies that the irrationality measure of the real number π k is µ ( π k ) = 2, see Definition10.1. Quod erat faciendum. (cid:4)
10 Basic Diophantine Approximations Results
All the materials covered in this section are standard results in the literature, see [13], [16], [19],[21], [24], [26], et alii.
Lemma 10.1.
Let α = [ a , a , . . . , a n , . . . , ] be the continue fraction of the real number α ∈ R .Then the following properties hold. (i) p n = a n p n − + p n − , p − = 0 , p − = 1 , for all n ≥ . (ii) q n = a n q n − + q n − , q − = 1 , q − = 0 , for all n ≥ . (iii) p n q n − − p n − q n = ( − n − , for all n ≥ . (iv) p n q n = a + X ≤ k
1, otherwise it is transcendental . Lemma 10.2.
If a real number α ∈ R is a rational number, then there exists a constant c = c ( α ) such that cq ≤ (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) (56) holds for any rational fraction p/q = α . Specifically, c ≥ /b if α = a/b . This is a statement about the lack of effective or good approximations for any arbitrary rationalnumber α ∈ Q by other rational numbers. On the other hand, irrational numbers α ∈ R − Q haveeffective approximations by rational numbers. If the complementary inequality | α − p/q | < c/q holds for infinitely many rational approximations p/q , then it already shows that the real number α ∈ R is irrational, so it is sufficient to prove the irrationality of real numbers. rrationality Exponents For Even Zeta Constants Lemma 10.3 (Dirichlet) . Suppose α ∈ R is an irrational number. Then there exists an infinitesequence of rational numbers p n /q n satisfying < (cid:12)(cid:12)(cid:12)(cid:12) α − p n q n (cid:12)(cid:12)(cid:12)(cid:12) < q n (57) for all integers n ∈ N . Lemma 10.4.
Let α = [ a , a , a , . . . ] be the continued fraction of a real number, and let { p n /q n : n ≥ } be the sequence of convergents. Then < (cid:12)(cid:12)(cid:12)(cid:12) α − p n q n (cid:12)(cid:12)(cid:12)(cid:12) < a n +1 q n (58) for all integers n ∈ N . This is standard in the literature, the proof appears in [13, Theorem 171], [24, Corollary 3.7], [14,Theorem 9], and similar references.
Lemma 10.5.
Let α = [ a , a , a , . . . ] be the continued fraction of a real number, and let { p n /q n : n ≥ } be the sequence of convergents. Then (i) 12 q n +1 q n ≤ (cid:12)(cid:12)(cid:12)(cid:12) α − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≤ q n , (ii) 12 a n +1 q n ≤ (cid:12)(cid:12)(cid:12)(cid:12) α − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≤ q n ,for all integers n ∈ N . The recursive relation q n +1 = a n +1 q n + q n − links the two inequalities. Confer [20, Theorem3.8], [14, Theorems 9 and 13], et alia. The proof of the best rational approximation stated below,appears in [21, Theorem 2.1], and [24, Theorem 3.8]. Lemma 10.6.
Let α ∈ R be an irrational real number, and let { p n /q n : n ≥ } be the sequence ofconvergents. Then, for any rational number p/q ∈ Q × , (i) | αq n − p n | ≤ | αq − p | , (ii) (cid:12)(cid:12)(cid:12)(cid:12) α − p n q n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ,for all sufficiently large n ∈ N such that q ≤ q n . The concept of measures of irrationality of real numbers is discussed in [26, p. 556], [5, Chapter11], et alii. This concept can be approached from several points of views.
Definition 10.1.
The irrationality measure µ ( α ) of a real number α ∈ R is the infimum of thesubset of real numbers µ ( α ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≪ q µ ( α ) (59)has finitely many rational solutions p and q . Equivalently, for any arbitrary small number ε > (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≫ q µ ( α )+ ε (60)for all large q ≥ Theorem 10.1. ([7, Theorem 2])
The map µ : R −→ [2 , ∞ ) ∪ { } is surjective function. Anynumber in the set [2 , ∞ ) ∪ { } is the irrationality measure of some irrational number. Example 10.1.
Some irrational numbers of various irrationality measures. rrationality Exponents For Even Zeta Constants µ ( α ) = 1, see [13, Theorem 186].(2) An algebraic irrational number has an irrationality measure of µ ( α ) = 2, an introduction tothe earlier proofs of Roth Theorem appears in [21, p. 147].(3) Any irrational number has an irrationality measure of µ ( α ) ≥ κ b = 0 . · · · b − · b · b + 1 · b + 2 · · · in base b ≥
2, concatenationof the b -base integers, has an irrationality measure of µ ( κ b ) = b . For example, the decimalnumber κ = 0 . · · · (61)has the irrationality measure of µ ( κ ) = 10.(5) A Mahler number ψ b = P n ≥ b − [ τ ] n in base b ≥ µ ( ψ b ) = τ ,for any real number τ ≥
2, see [7, Theorem 2]. For example, the decimal number ψ = 110 + 110 + 110 + 110 + · · · (62)has the irrationality measure of µ ( ψ ) = 3.(6) A Liouville number ℓ b = P n ≥ b − n ! parameterized by b ≥ µ ( ℓ b ) = ∞ , see [13, p. 208]. For example, the decimal number ℓ = 110 + 110 + 110 + 110 + · · · (63)has the irrationality measure of µ ( ℓ ) = ∞ . Definition 10.2.
A measure of irrationality µ ( α ) ≥ α ∈ R × is amap ψ : N → R such that for any p, q ∈ N with q ≥ q , (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ ψ ( q ) . (64)Furthermore, any measure of irrationality of an irrational real number satisfies ψ ( q ) ≥ √ q µ ( α ) ≥√ q . Theorem 10.2.
For all integers p, q ∈ N , and q ≥ q , the number π satisfies the rational approx-imation inequality (cid:12)(cid:12)(cid:12)(cid:12) π − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ q . . (65) Proof.
Consult the original source [25, Theorem 1]. (cid:4)
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