for ,
where
is the gamma function and is the Riemann zeta
function (although care must be taken at because of the removable
singularity present there). It was conjectured by Hadjicostas (2004) and almost
immediately proved by Chapman (2004). The special case gives Beukers's integral for ,
(3)
(Beukers 1979). At ,
the formula is related to Beukers's integral for Apéry's
constant ,
which is how interest in this class of integrals originally arose.
There is an analogous formula
(4)
for ,
due to Sondow (2005), where is the Dirichlet
eta function. This includes the special cases
Beukers, F. "A Note on the Irrationality of and ." Bull. London Math. Soc.11, 268-272,
1979.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation
in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters,
2004.Chapman, R. "A Proof of Hadjicostas's Conjecture." 15
Jun 2004. http://arxiv.org/abs/math/0405478.Guillera,
J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical
Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005
http://arxiv.org/abs/math.NT/0506319.Hadjicostas,
P. "A Conjecture-Generalization of Sondow's Formula." 21 May 2004. http://www.arxiv.org/abs/math.NT/0405423/.Sloane,
N. J. A. Sequences A094640, A103130
in "The On-Line Encyclopedia of Integer Sequences."Sondow,
J. "Criteria for Irrationality of Euler's Constant." Proc. Amer. Math.
Soc.131, 3335-3344, 2003. http://arxiv.org/abs/math.NT/0209070.Sondow,
J. "Double Integrals for Euler's Constant and and an Analog of Hadjicostas's Formula." Amer.
Math. Monthly112, 61-65, 2005.