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. (Chakraborty et al [CKW]) For non-integral values of z, we have the partial fraction expansion ∞   πz= 1 + 1 1 + 1 cot πz π z − n z + n n=1 ∞ 1 2z  1 = + (2.1) πz π z2 − n2 n=1 which is equivalent to the functional equation (1.4) for the Riemann zeta-function. Also we need a generalization of (1.7).

LEMMA 1. We have  z sin πz G(1 + z) log sin πt dt = z log + log ,z/∈ Z. (2.2) 0 2π G(1 − z) Proof. We take the following generic formula for granted (which is a form of the Weierstrass product representation [SC, (3), p. 25]): ∞   G  n z 1 (1 + z) = − 1 + + (log 2π − 1) − (1 + γ)z. (2.3) G z + n n 2 n=1 We compare (2.3) and (2.1). From (2.3) we obtain   G G ∞ ( − z) + ( + z) = n 1 + 1 − + π − . G 1 G 1 n + z n − z 2 log 2 1 n=1 Rewriting the summand   1 1 z2 n + − 2 = 2 , n + z n − z n2 − z2 we see that G G ( − z) + ( + z) =−πz πz+ π. G 1 G 1 cot log 2 (2.4) On integrating, we conclude that  z G(1 − z) πt cot πt dt = log + z log 2π, (2.5) 0 G(1 + z) which is due to Kinkelin and is stated as [SC, (26), p. 30]. By integration by parts, we may deduce (2.2) from (2.5). 2 The following lemma is needed in the proof of Theorem 2.

LEMMA 2. We have   G( 3 ) 1 G = 2π log 4 + log 2π2 . (2.6) (1 )G( 1 ) 8 4 4 110 X. Wang

Proof. A special case of (2.2) reads  π/4 √ 5 − 1 G( ) π 1 log sin tdt=− log 2π + log 4 . (2.7) 4 G( 3 ) 0 4 Comparing (1.7) and (2.7), we obtain

G( 5 ) π π 1 π log 4 = log 2 + log π − G, G( 3 ) 8 4 2 4 which is (2.6). 2

Equation (2.6) may be further simplified if we appeal to the duplication formula (2.17) for G.

LEMMA 3. We have  z/2 z π 1 cos(π/2)z log tan πt dt = log tan z + log 2 2 2 π 0   1 − z G(1 + z/2) G((1 + z)/2) + log  + log − log . (2.8) 2 G(1 − z/2) G((1 − z)/2) Proof. Using (1.14), we may rewrite (2.2) in the form    z 1 cos πz log cos πt dt = z − log 0 2 2π   1 1 G( 1 + z) − log 2 − log  − z + log 2 . (2.9) 2 2 G( 1 − z) 2 Combining (2.2) and (2.9), and replacing z by z/2, we deduce (2.8). 2

The proof of Theorem 3 can be given by applying, for example, a theorem of Titchmarsh [T, p. 110].

PROPOSITION 1. (Titchmarsh [T]) Suppose that f(z)is a meromorphic function with simple pole at ak, k ∈ N with 0 < |a1|≤|a2|≤··· with residue bk = Resz=ak F(z). Suppose that there is a sequence of closed contours CN ,N∈ N, including a1,...,aN only, such that the minimum distance RN of CN from the origin tends to ∞ as N →∞while the circumference LN of CN is O(RN ) and such that f(z)= o(RN ) on CN . Under these conditions ∞    1 1 f(z)= f(0) + bn + (2.10) z − an an n=1 for all values of z except for poles.

We may deduce the following as a consequence of Proposition 1: ∞ 1 1 2  (−1)nz = + . (2.11) sin πz πz π z2 − n2 n=1 The Barnes G-function and the Catalan constant 111

LEMMA 4. We have the Maclaurin expansion ∞ 1−2n t (1 − 2 )ζ(2n) n = 1 + 2 t2 , |t| <π. (2.12) sin t π2n n=1 Proof. We deduce (2.12) from the partial fraction expansion (2.11). For |t| <π note that ∞ t  (−1)n 1 = 1 − 2 t2 . sin t (πn)2 1 − (t/πn)2 n=1  ∞ (t/πn)2(m−1) Substituting the geometric series m=1 and changing the order of summation, we obtain ∞ ∞ n t (−1) m = 1 − 2 t2 . (2.13) sin t (πn)2m m=1 n=1 Recalling that ∞ (− )n 1 = ( − 1−s)ζ(s), ns 1 2 (2.14) n=1 we deduce (2.12) from (2.14). 2 In the second statement of Theorem 2 we use the formula [GR, 3. 612.5].

LEMMA 5. We have  n− 1 sin 2πnt 4(−1)n+1 1 (−1)k dt = . (2.15) cos πt π 2k + 1 0 k=0 The proof immediately follows by expressing the trigonometric functions in terms of complex exponential functions and applying the geometric series.

LEMMA 6. The Fourier series expansion ∞ πnx B¯ (x) =−1 sin 2 ,x/∈ Z 1 π n (2.16) n=1 is a consequence of the functional equation (1.4). Proof of Theorem 1. We use (2.8) and the duplication formula for the G-function [SC, (12), p. 30]   2 1 z − z2 G(z)G z + G(z + 1) = κ(2)(8π) 2 2 G(2z) (2.17) 2 in the form       z z + 2 z 1 1 z −z2/2 G G G 1 + = κ(2)(8π)2 2 G(z), (2.18) 2 2 2 where κ(2) is a constant, to deduce (1.9). Since   z πt z/2 2πu S(z) = dt = 2 du 0 sin πt 0 sin 2πu  z/2 =[ u πu]z/2 − πu du, 2 log tan 0 2 log tan 0 112 X. Wang it follows, on substituting (2.8), that   cos(π/2)z 1 − z S(z) =−log − 2log π 2 G(1 + z/2) G((1 + z)/2) − 2log + 2log . (2.19) G(1 − z/2) G((1 − z)/2) We want to eliminate the last term on the right-hand side of (2.19). We slightly transform (2.18) by multiplying it by     −1 z −z√ 1 + z  = 21 π(z) , 2 2 which is the duplication formula for the gamma function:       2 −1 z 1 + z z/ −z2/ √ 1 + z G 1 + G = κ(2)(2π) 22 2 π G(1 + z), 2 2 2 by (1.14). Hence,     z 1 + z z −z2/ √ 2logG 1 + G = log κ(2) + log 2π + log 2 2 π 2 2 2   1 + z − log  + log G(1 + z), 2 and     z 1 − z z −z2/ √ 2logG 1 − G = log κ(2) − log 2π + log 2 2 π 2 2 2   1 − z − log  + log G(1 − z). 2 Hence, G((1 + z)/2) ((1 + z)/2) G(1 + z) G(1 + z/2) 2log = z log 2π − log + log − 2log . G((1 − z)/2) ((1 − z)/2) G(1 − z) G(1 − z/2) Substituting this into (2.19), we obtain   cos(π/2)z 1 − z ((1 + z)/2) S(z) = z log 2π − log − 2log − log π 2 ((1 − z)/2) G(1 + z) G(1 + z/2) + log − 4log . G(1 − z) G(1 − z/2) This amounts to formula (1.9) in view of the reciprocal relation for the gamma function ((z)(1 − z) = π/sin πz), completing the proof. 2 z = 1 Proof of Theorem 2. Putting 2 in (1.9), we have   1 1 G( 3 ) G( 5 ) S = log 2π + log 2 − 4log 4 . 2 2 G( 1 ) G( 3 ) 2 4 The Barnes G-function and the Catalan constant 113

Since G( 3 ) (1 )G( 1 ) √ log 2 = log 2 2 = log π, G( 1 ) G( 1 ) 2 2 and (1 )G( 1 ) G( 5 ) 1 1 G −log 4 4 =−log 4 =− log 2 − log π + G( 3 ) G( 3 ) 8 4 2π 4 4 by (2.6), it follows that   G S 1 = 2 , 2 π which is the same as (1.10), and the first half of Theorem 2 is proved. Now we deduce (1.11) from the Fourier series (2.16). Since the series in (2.16) is uniformly convergent, we obtain   1 B (t) ∞ 1 πnt − 1 dt = 1 sin 2 dt, cos πt π cos πt 0 n=1 0 whose right-hand side is the right-hand side of (1.11) up to a constant factor by (2.15). It therefore remains to prove that the left-hand side is (4/π2)G. By the change of variable, we see that   1 B (t) π/2 t − 1 dt = 2 dt, 2 0 cos πt π 0 sin t whence (1.11) follows in view of (1.10). 2 Another proof of (1.10). We are to prove that   1 πS = 2G. (2.20) 2 By (2.12),   ∞   n+ 1 π  (1 − 21−2n)ζ(2n) 1 π 2 1 πS = + 2 (2.21) n + π2n 2 2 n= 2 1 2 1    π 1 1 = + πF − 2πF , 2 2 4 say, where ∞ ζ(2n) n F(z)= z2 , |z| < 1. (2.22) 2n + 1 n=1 Since ∞ ζ(n) n n 2F(z)= (z + (−z) ), n + 1 n=2 the bisection formula, it follows from (1.18) with α = 1that

2F(z)= ζ (0, 1 − z) + ζ (0, 1 + z)

+ (ζ (− , − z) − ζ (− , + z))1 − ζ( ). 1 1 1 1 z 2 0 (2.23) 114 X. Wang

Recalling Lerch’s formula (z) ζ (0,z)= log √ , (2.24) 2π we find as in the last lines of proof of Theorem 1 that the first two terms on the right-hand side of (2.23) are log(z/2sinπz). To transform the coefficient ζ (−1, 1 − z) − ζ (−1, 1 + z) of the third term on the right of (2.24) we introduce the function Ak(q) (see [EM2]),

Ak(q) = kζ (1 − k, q), (2.25) which is closely related to the polygamma function of negative order, and specifically

A2(q) = 2ζ (−1,q). Hence, (2.23) can be written as z 1 2F(z)= log + (A˜ (1 − z) − A˜ (1 + z)) − 2ζ(0). (2.26) 2sinπz 2z 2 2 Substituting (2.26) in (2.21), we obtain         1 π π 1 1 3 πS = + log 2 + A˜ − A˜ + 1 (2.27) 2 2 2 2sin(π/2) 2 2 2 2        1 3 5 − π log 4 + 2 A˜ − A˜ + 1 . 2sin(π/4) 2 4 2 4 ˜ We recall some properties of Ak(q): ˜ ˜ A2(q + 1) = A2(q) + 2q log q, (2.28)   1 1 A˜ =−ζ (−1) − log 2, (2.29) 2 2 12   1 1 G A˜ =− ζ (−1) + . (2.30) 2 4 4 2π

A ( 3 ) We need one more formula for 2 4 , which can be read off from the distribution property   4 a s ζ s, = 4 ζ(s). 4 a=1 By differentiation and setting s =−1,     3 1 1 A˜ =− ζ (−1) − A˜ . (2.31) 2 4 2 2 4 Substituting (2.28), (2.31) in (2.27), we conclude that       1 1 1 πS = 2π 2A + ζ (−1) , 2 2 4 2 amounting to (1.10). This complete the proof. 2 The Barnes G-function and the Catalan constant 115 ˜ Remark 1. The relation between those special functions and A2(x) is given by

1 1 1  (x) A˜ (x) − = ζ (−1,x)− = log 2 + (x − 1) log (x) 2 2 12 12 A

=−log AG(x) + (x − 1) log (x) = A2(x) (2.32) (see [SC, (83), p. 94]). Using (2.24), etc., in (2.23), we deduce that 1 G(1 + z) z zF(z) = log − log (1 − z)(1 + z) 2 G(1 − z) 2 z z + (log (1 − z)(1 + z) − log 2π)+ , 2 2 i.e. ∞ ζ(2n) n+ 1 1 G(1 + z) z2 1 = (1 − log 2π)z+ log , |z| < 1 2n + 1 2 2 G(1 − z) n=1 which is [SC, (464), p. 212]. Remark 2. Compared with (2.21), it might be interesting to evaluate the ‘rational series’ n−1 2n which follows by substituting ζ(2n) = (−1) (B2n/(2n)!)π :    ∞ 2n 1 2 − 2 n− B n t 1 + (−1) 1 2 = dt . (2.33) 2n + 1 (2n)! sin t n=1 0 Along with (1.6), the following evaluation is also well known  ∞ log t dt = 0. (2.34) 2 0 1 + t Equations (2.34) and (1.6) are due to Euler and constitute the examples of improper integrals for which the indefinite integral cannot be expressed in terms of elementary functions, but which still can be evaluated by some ingenious ways in the framework of new transcendental functions. In connection with (2.34), the evaluation [SC, p. 33]  1 t log dt =−G 2 (2.35) 0 1 + t is of some interest. This can be easily deduced using formula (2.8). By the change of variable t ←→ 1/π arctan at, equation (2.8) leads to [SC, (46), p. 33]    z log t log z π cos arctan ax dt = arctan ax + log 2 2 0 1 + a t a 2a π π G(1 + (1/π) arctan ax)G(3 − (1/π) arctan ax) + log 2 , (2.36) a G( − ( /π) ax)G(1 + ( /π) ax) 1 1 arctan 2 1 arctan whose special case with a = z = 1 reads  1 log t π 1 G( 5 )G( 5 ) dt = log √ + π log 4 4 1 + t2 2 π G( 3 )G( 3 ) 0 2 4 4 which leads to (2.35) on account of (2.6). 116 X. Wang

There are many instances where some complicated integrals or series may be expressed in terms of the G-functions; for example, for a/∈ Z we have  a 1 1 log (1 − t) dt = log G(1 − a) + a log (1 − a) + a2 + (log 2π − 1)a, (2.37) 0 2 2 which is a corrected version of [SC, (13), p. 240]. We have hopefully taken one step forward toward the understanding of special values of the L-function in the opposite parity case as being intrinsic to an appropriate generalization of the gamma function. Also the relation between intrinsicity of G to the double gama function and (1.13) has strong resemblance to the celebrated Chowla–Selberg formula which exhibits a similar relation between the values of the gamma and the elliptic function. We hope to return to the study of this type of problem on another occasion.

Acknowledgements. This paper forms part of the author’s doctoral thesis to be submitted to Kinki University. The author would like to express her hearty thanks to Professor Shigeru Kanemitsu for thorough supervision and encouragement. Thanks are also due to Professor Hailong Li for his generosity in allowing the author to use part of the material in his book [Li] before publication.

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Xiaohan Wang Xi’an International Studies University Xi’an, Shaanxi 710128 China (E-mail: [email protected])