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Irrationality of the Zeta Constants Ζ(2N) N 1

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Irrationality of the Zeta Constants Ζ(2N) N 1 Irrationality of the Zeta Constants N. A. Carella Abstract: A general technique for proving the irrationality of the zeta constants ζ(2n + 1),n 1, ≥ from the known irrationality of the beta constants L(2n + 1,χ) is developed in this note. The irrationality of the zeta constants ζ(2n),n 1, and ζ(3) are well known, but the irrationality ≥ results for the zeta constants ζ(2n + 1),n 2, are new, and seem to show that these are irrational ≥ numbers. By symmetry, the irrationality of the beta constants L(2n,χ) are derived from the known rrationality of the zeta constants ζ(2n). Keyword: Irrational number, Transcendental number, Beta constant, Zeta constant, Uniform dis- tribution. AMS Mathematical Subjects Classification: 11J72; 11A55. 1 Introduction As the rationality or irrationality nature of a number is an arithmetic property, it is not surprising to encounter important constants, whose rationality or irrationality is linked to the properties of the in- tegers and the distribution of the prime numbers, for example, the number 6/π2 = 1 p−2 . p≥2 − An estimate of the partial sum of the Dedekind zeta function of quadratic numbers fields will be Q arXiv:1212.4082v7 [math.GM] 22 Jun 2018 utilized to develop a general technique for proving the irrationality of the zeta constants ζ(2n + 1) from the known irrationality of the beta constants L(2n+1,χ), 1 = n N. This technique provides 6 ∈ another proof of the first odd case ζ(3), which have well known proofs of irrationalities, see [2], [7], [34], et al, and an original proof for the other odd cases ζ(2n + 1),n 2, which seems to confirm ≥ the irrationality of these number. Theorem 1.1. For each fixed odd integer s = 2k + 1 3, the zeta constant ζ(s) is an irrational ≥ number. The current research literature on the zeta constants ζ(2n + 1) states the following: 1. The special zeta value ζ(3) is an irrational number, see [2], [7], [18], [34], et al. 2. At least one of the four numbers ζ(5),ζ(7),ζ(9), and ζ(11) is an irrational number, see [39, p. 7] and 1 zeta constants and beta constants [42]. 3. The sequence ζ(5),ζ(7),ζ(9),ζ(11),... contains infinitely many irrational zeta constants, see [8]. Various advanced techniques for studying the zeta constants are surveyed in [28], and [37]. By the symmetry of the factorization of the Dedekind zeta function ζK(s) = ζ(s)L(s,χ) with re- spect to either ζ(s) or L(s,χ), almost the same analysis leads to a derivation of the irrationality of the beta constants L(2n,χ) from the known irrationality of the zeta constants ζ(2n) n 1. ≥ Theorem 1.2. For each fixed even integer s = 2n 2, and the nonprincipal quadratic character ∞ −s≥ χ mod 4, the beta constant L(s,χ)= n=1 χ(n)n is an irrational number. Section 2 contains basic materials onP the theory of irrationality, the transcendental properties of real numbers, and an estimate of the summatory function of the Dedekind zeta function. The proof of Theorem 1.1 is given in section 6. Last but not least, a sketch of the proof of Theorem 1.2 is given in the last Section. 2 Fundamental Concepts and Background The basic notation, concepts and results employed throughout this work are stated in this Section. All the materials covered in this subsection are standard definitions and results in the literature, confer [19], [24], [29], [34], [39], et al. 2.1 Criteria for Rationality and for Irrationality A real number α R is called rational if α = a/b, where a, b Z are integers. Otherwise, the ∈ ∈ number is irrational. The irrational numbers are further classified as algebraic if α is the root of an irreducible polynomial f(x) Z[x] of degree deg(f) > 1, otherwise it is transcendental. ∈ Lemma 2.1. (Criterion for rationality) If a real number α Q is a rational number, then there ∈ exists a constant c = c(α) such that c p α (1) q ≤ − q holds for any rational fraction p/q = α. Specifically, c 1/B if α = A/B. 6 ≥ This is a statement about the lack of effective or good approximations of an arbitrary rational number α Q by other rational numbers. On the other hand, irrational numbers α R Q have ∈ ∈ − effective 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 almost irrational, so it is almost sufficient ∈ to prove the irrationality of real numbers. Lemma 2.2. (Criterion for irrationality) Let ψ(x) = o(1/x) be a monotonic decresing function, and let α Q be a real number. If ∈ p 0 < α < ψ(q) (2) − q 2 zeta constants and beta constants holds for infinitely many rational fraction p/q Q, then α is irrational. ∈ Proof. By Lemma 2.1 and the hypothesis, it follows that c p 1 α < ψ(q)= o . (3) q ≤ − q q But this is a contradiction since c/q = o(1/q). 6 More precise results for testing the irrationality of an arbitrary real number are stated below. Theorem 2.1. (Dirichlet) Suppose α R is an irrational number. Then there exists an infinite ∈ sequence of rational numbers pn/qn satisfying p 1 0 < α n < (4) − q q2 n n for all integers n N. ∈ For continued fractions α = [a , a , a ,...] with sizable entries a a > 1, where a ia a constant, 0 1 2 i ≥ there is a slightly better inequality. Theorem 2.2. Let α = [a , a , a ,...] be the continued fraction of a real number, and let p /q : 0 1 2 { n n n 1 be the sequence of convergents. Then ≥ } p 1 0 < α n < (5) − q a q2 n n n for all integers n N. ∈ This is standard in the literature, the proof appears in [19, Theorem 171], [34, Corollary 3.7], and similar references. A more general result provides a family of inequalities for almost all real numbers. Theorem 2.3. (Khinchin) Let ψ be a real and deceasing function, and let α R be a real number. ∈ If there exists an infinite sequence of rational approximations p /q such that p /q = α, and n n n n 6 p ψ(q ) α n < n (6) − q q n n and ψ(q) < , then real number α is ψ-approximatable. q ∞ P 3 Estimate of An Arithmetic Function Let q 2 be an integer, and let χ = 1 be the quadratic character modulo q. The Dedekind zeta ≥ 6 function of a quadratic numbers field Q √q is defined by ζK(s)= ζ(s)L(s,χ). The factorization consists of the zeta function ζ(s) = ∞ n−s, and the L-function L(s,χ) = ∞ χ(n)n−s, see n=1 n=1 [11, p. 219]. The product has the Dirichlet series expansion P P ∞ ∞ ∞ 1 1 1 ζ(s) L(s,χ)= χ(n) = χ(d) , (7) · ns ns ns n=1 ! n=1 ! n=1 X X X Xd|n 3 zeta constants and beta constants and its summatory function is n≤x r(n) = 4 n≤x d|n χ(d), see [21, p. 17]. P P P The finite sum rQ(n) = 4 d|n χ(d) is the number of respresentations of n by primitive forms Q(u, v)= au2 + buv + cv2 of discriminant q. For q = 4, the nonprincipal character χ = 1 has order P 6 2, and the counting function r(n) = 4 χ(d)=# (u, v) : n = u2 + v2 0 tallies the number d|n ≥ of representations of an integer n 1 as sums of two squares. ≥ P Lemma 3.1. The average order of the summatory function of the Dedekind function is given by r (n) c 1 Q = ζ(s)L(s,χ)+ 0 + O , (8) ns xs−1 xs−1/2 nX≤x where c > 0 is a constant, and x 1 is a large number. 0 ≥ A proof appears in [25, p. 369], slightly different version, based on the hyperbola method, is computed in [32, p. 255]. 4 The Irrationality of Some Constants The different analytical techniques utilized to confirm the irrationality, transcendence, and ir- rationality measures of many constants are important in the development of other irrationality proofs. Some of these results will be used later on. Theorem 4.1. The real numbers π, ζ(2), and ζ(3) are irrational numbers. The various irrationality proofs of these numbers are widely available in the open literature. These technique are valuable tools in the theory of irrational numbers, refer to [2], [7], [18], [34], and others. Theorem 4.2. For any fixed n N, and the nonprincipal character χ mod 4, the followings ∈ statements are valid. ( 1)n+122nB (i) The real number ζ(2n)= − 2n π2n is a transcendental number, (2n)! ( 1)nE (ii) The real number L(2n + 1,χ)= − 2n π2n+1 is a transcendental number, 22n+2(2n)! where B2n and E2n are the Bernoulli and Euler numbers respectively. Proof. Apply the Lindemann-Weierstrass theorem to the transcendental number π. The first few nonvanishing Bernoulli numbers and Euler numbers are as these. −1 1 −1 1 1. B0 = 1,B1 = 2 ,B2 = 6 ,B4 = 30 ,B6 = 42 ,..., 2. E = 1, E = 1, E = 5, E = 161,... 0 2 − 4 6 − The generalization of these results to number fields is discussed in [40], and related literature.
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