Series

Páginas: 10 (2496 palabras) Publicado: 28 de septiembre de 2011
Comm. Korean Math. Soc. 131998, No. 2, pp. 435-443

THE CATALAN'S CONSTANT AND SERIES INVOLVING THE ZETA FUNCTION
Junesang Choi
Abstract. Some mathematical constants have been used in evaluating series involving the Zeta function, the origin of which can be traced back to an over two centuries old theorem of Christian Goldbach. We show some of the series involving the Zeta function can beevaluated in terms of the Catalan's constant by using the theory of the double Gamma function.

1. Introduction and de nitions The Catalan's constant G is de ned by
1.1
1 1 Z 1 Kkdk = X ,1m  0:915965 : : : ; G := 2 2m + 12 = 0 m=0
Z

where K is the complete elliptic integral of the rst kind given by

dt : 0 1 , k2 sin2 t The double Gamma function was de ned and studied by Barnes35 and others in about 1900, not appearing in the tables of the most well-known special functions, but cited in the exercise by Whittaker and Watson 23, p. 264 . Recently this function has been revived in the study of determinants of Laplacians 8, 15, 21, 22 . Shintani 17 also used this function to prove the classical Kronecker limit formula. Its p-adic analytic extension appeared in a formula ofCassou-Nogu s 7 for e the p-adic L-functions at the point 0.

Kk :=

=2

p

Received July 3, 1997. Revised January 13, 1998. 1991 Mathematics Subject Classi cation: Primary 33B99, Secondary 11M99. Key words and phrases: the Catalan's constant, series involving the Zeta function.

436

Junesang Choi

Before Barnes, these functions had been introduced under a di erent form byAlexeiewsky 2 , Holder 12 and Kinkelin 14 . Barnes 3 gave an explicit Weierstrass canonical product form of the double Gamma function ,2 = 1=G : 1.2

f,2 z + 1g,1 = Gz + 1
z

2 = 2 2 e, 1

1+  2 +

z z

1 Y

k=1

2 z k 2 1 + k e,z+ zk ;

where is the Euler-Mascheroni constant de ned by 1.3 1 , log n  0:577215664 : : : : = nlim = !1 k=1 k
 n X !

The double Gammafunction and the Gamma function satisfy the following relations: ,1 = 1 and G1 = 1 1.4 ,z + 1 = z,z and Gz + 1 = ,zGz for z 2 C; where , is the well-known Gamma function whose Weierstrass canonical product form is 1.5
1
z ,1 = ze z Y 1 + z e, k : f,zg k k=1

The Stirling's formula for the G-function: For su ciently large real x and a 2 C; 2 1 log Gx + a + 1 = x + a log2, log A + 12 , 3x , ax 2 4 2
1.6 2 1 + x , 12 + a + ax log x + O1=x; 2 2 where A is Glaisher's or Kinkelin's constant de ned by 2
2 1 2 n  , n + n + 1 log n + n ; 1.7 log A = nlim log1 2    n !1 2 2 12 4

The Catalan's constant and series involving the Zeta function

437

the numerical value of A being 1:282427130    : It is also known that 3

1 =  2 andG 1 = 2 24 , 1 e 1 A, 3 : 1 1 4 8 2 1.8 , 2 2 Note 1, p. 189, Eq. 11 that the partial fraction expansion for z cot z is given by 1.9

z cot z = 1 + 2

z2 ; 2 2 n=1 z , n

1 X

and we nd 1, p. 199, Eq. 30 that 1.10

The Riemann Zeta function
s is de ned, when Res 1, by see Titchmarsh 20 and Ivi
13  c 1.11
1 1 = 1 X 1 :
s = ns 1 , 2,s 2n , 1s n=1 n=1 1X

,z,1 , z = sinz :

Indeed it is meromorphic everywhere in the complex s-plane with a simple pole at s = 1 with residue 1. More recently Choi et al.  10 , 11  showed that the theory of the double Gamma function turned out to be useful in evaluating some series involving the Zeta function, the origin of which can be traced back to an over two centuries old theorem of ChristianGoldbach 1690-1764 as noted in Srivastava 19 . The Kinkelin's constant A was used in evaluating some series involving the Zeta function: For example, 1.12
1 X
k=2

,1k
k , 1 = , 1 log

k+2

2



+ 3 , 1 + 2 log A; 2 3

which was shown by Choi et al. 10, p. 391, Eq. 2.31 ; 1.13
1 X

k 2,k = , + 7 log 2 + 1 log  , 3 log A; 4 12 2 k=2 kk + 1

438

Junesang...
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