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The Barnes G-Function and the Catalan Constant

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The Barnes G-Function and the Catalan Constant Kyushu J. Math. 67 (2013), 105–116 doi:10.2206/kyushujm.67.105 THE BARNES G-FUNCTION AND THE CATALAN CONSTANT Xiaohan WANG (Received 24 December 2011 and revised 14 February 2012) Abstract. G = ∞ [(− )k/( n − )2] The Catalan constant, n=1 1 2 1 , has attracted much attention and has appeared in various contexts. It is the special value at s = 2oftheDirichletL- function associated with the Gaussian field with its character odd. In this paper, we appeal to the Barnes G-function and a form of the functional equation for the Riemann zeta-function, to deduce an expression for the integral in t/sin t and one for G. These give rise to a corollary which reveals the intrinsicity of the Catalan constant G to the Barnes G-function. We also use one of Ramanujan’s formulas to derive a theorem. 1. Introduction and statement of results Let ∞ ζ(s, a)= 1 , s = σ> (n + a)s Re 1 (1.1) n=0 denote the Hurwitz zeta-function whose special case is the Riemann zeta-function ∞ ζ(s)= ζ(s, ) = 1 ,σ>. 1 ns 1 (1.2) n=1 The Dirichlet L-function L(s, χ) associated with the Dirichlet character χ is defined by ∞ χ(n) L(s, χ) = ,σ>. ns 1 (1.3) n=1 The parity of χ is defined by the parity of a in χ(−1) = (−1)a and χ is called even or odd according to whether a is even or odd. Hence, − χ(− ) 0 χ even a = a(χ) = 1 1 = 2 1 χ odd. L χ∗ We may view the Riemann zeta-function as an -function with the even character 0 whose value is 1 for all values of n = 0. We recall that these zeta-and L-functions satisfy functional equations and, in particular, −s/ s −( −s)/ 1 − s π 2 ζ(s)= π 1 2 ζ(1 − s), (1.4) 2 2 2010 Mathematics Subject Classification: Primary 11F66, 11M26, 11M41. Keywords: Barnes G-function; Catalan constant; functional equation; Ramanujan’s formula. c 2013 Faculty of Mathematics, Kyushu University 106 X. Wang χ∗ χ q which is the special case 0 of the following. For a primitive Dirichlet character modulo we have √ ia q ξ( − s, χ) = ξ(s, χ), 1 τ(χ) where s + a ξ(s, χ) = π−(s+a)/2 L(s, χ), 2 τ(χ)= χ(a)e2πa/q q and a mod q is the normalized Gauss sum to the modulus . The Catalan constant, denoted by G, is our main concern and is defined by the absolutely convergent series ∞ (−1)n−1 G = = L(2,χ4), (1.5) (2n − 1)2 n=1 where χ4 is the non-principal Dirichlet character mod 4. Its value is computed to be G 0.915695594177219015 ··· (see [SC, (16), p. 29]). The interest of this seemingly simple-looking constant lies rather deep and can be χ perceived in the following√ context. The character 4 is the Kronecker character associated with the Gaussian field Q( −1) which is the imaginary quadratic field and its character is odd. In general, the special values of the L-function at integral arguments, say k ≥ 0, have 2 2 been studied with great interest. It is well known that ζ(2) = π /6 = B2π , which is the same parity case. As is well known, in the same parity case, the special values are classically known in terms of (generalized) Bernoulli numbers. The k = 1 case is known as the Dirichlet class number formula and the case χd being even (real quadratic case) is the more complicated case, where the Clausen function appears in place of Bernoulli polynomials. The Catalan constant is the second simplest opposite parity case. The interest in the Catalan constant also comes from another context involving the Barnes G-function. The Euler integral π/2 π log sin xdx=− log 2 (1.6) 0 2 is well known and is used in generalizing the Gauss mean value theorem in [A]. In this connection, the formula [GR, p. 526, Entry 4.224] (see also [SC, (18), p. 29]) π/4 π 1 log sin xdx=− log 2 − G (1.7) 0 4 2 is noteworthy and we may expect similar formulas to hold. Since G is a special value of the Dirichlet L-function which satisfies the functional equation, it is natural to interpret (1.7) as a manifestation of the functional equation. The answer is given by Theorem 1 which gives an expression for a sine integral in terms of the Barnes G-function. The proof depends on the defining properties of the G-function and the partial fraction expansion (2.1) for the cotangent function, the latter of which is equivalent to the functional equation for the The Barnes G-function and the Catalan constant 107 Riemann zeta-function up to a Mellin transform formula (Theorem 3). Further, combining the special case of (1.9) with the Fourier series for the first periodic Bernoulli polynomial (which is also a consequence of the functional equation), we deduce another expression (1.10) for G (Theorem 2). These two theorems combined imply Corollary 1 to the effect that the functional equation, as symmetry s ↔ 1 − s, works as catalysis and disappears at the end, which suggests that G is a constant intrinsic to the Barnes G-function in the same way as the Euler constant γ is with the gamma function. This is in conformity with our intuition that the special values of the L-function in the opposite parity case are independent of the functional equation (the corresponding values at odd integer arguments being zero). Further, using one of Ramanujan’s formulas (1.18), we give another proof of (1.10) (Theorem 2). This would suggest that Ramanujan’s formula might be independent of the functional equation. For the moment we interpret the Barnes G-function as a relative of the gamma function, −1 2 , in the class of multiple gamma functions and we state the precise definition later (cf. [B1, B2]). Our main results are the following two theorems. THEOREM 1. Let z πt 1 πz t S(z) = dt = dt (1.8) 0 sin πt π 0 sin t once and for all. Then except for integer values of z, we have G(1 + z) G(1 + z/2) S(z) = z log 2π + log − 4log , (1.9) G(1 − z) G(1 − z/2) as a consequence of the functional equation (1.4). THEOREM 2. The evaluation 1 π/2 t πS = dt = 2G (1.10) 2 0 sin t is a consequence of (1.9) and the functional equation (1.4). It is also a consequence of Ramanujan’s formula (1.18) as well as (1.9). Theorems 1 and 2 when combined give rise to (1.11). COROLLARY 1. The expression for the Catalan constant ∞ n− (−1)n−1 1 (−1)k G = (1.11) n 2k + 1 n=1 k=0 is a consequence of the defining properties of the G-function, and a fortiori, G is intrinsic to the Barnes G-function. Equation (1.11) is [EM1, (8.6), p. 178]. Proofs are given in Section 2. The Catalan constant appears in many other contexts and we state an expression for it for curiosity (cf. also Remark 2 at the end of the paper). Let K = K(k) be the complete elliptic integral of the first kind defined by 1 dt π/2 dt K(k) = = , |k| < 1 (1.12) 0 (1 − t2)(1 − k2t2) 0 1 − k2 sin2 t 108 X. Wang (see [SC, (17), p. 29]), where k is called the modulus. It is known that 1 1 G = K(k) dk. (1.13) 2 0 Now we give the definition of the Barnes G-function and state some of its properties. The Barnes G-function is defined as the solution to the difference equation log G(z + 1) − log G(z) = log (z) (1.14) with the initial condition log G(1) = 0 and the asymptotic formula to be satisfied: N + 1 + z log G(z + N + 2) = log 2π 2 1 1 + N2 + 2N + 1 + + z2 + 2(N + 1)z log N 2 6 3 1 − − N2 − N − Nz − log A + + O(N 1), N →∞, (1.15) 4 12 where log A is the Glaisher–Kinkelin constant defined by N log A = lim (n + a) log(n + a) N→∞ n= 1 1 1 a − (N2 + (2a + 1)N + a2 + a + B ) log(N + a) + N2 + N 2 2 4 2 (1.16) for a>0. There are many formulas known that correspond to those of the gamma function (cf. (2.3)) and 1 log G(z) =− ζ (−1,z)− − log A − (1 − z) log (z), z∈ / Z. (1.17) 12 Now we recall Ramanujan’s formula (1.18) alluded to above in its general case, which will turn out to be useful in this type of evaluation (cf. e.g. [KT, Ch. 3]). For 0 ≤ λ ∈ Z and |z| < |α| we have ∞ ζ(n,α) λ λ zn+λ = ζ (−k, α − z)zλ−k − ζ (−λ, α) n + λ k n=2 k=0 λ 1 k 1 λ+1 − ζ(k − λ, α)z + (ψ(α) − Hλ)z , (1.18) k λ + 1 k=1 where ψ indicates the digamma function, Hλ is the λth harmonic number, and where ψ(1) =−γ . The Barnes G-function and the Catalan constant 109 2. Proofs of results The following theorem constitutes a fundamental ingredient of the proofs of theorems. THEOREM 3.
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