In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the Gamma function and the Riemann zeta function.

## Statement

Let s be a complex number with s ≠ -1 and Re(s) > −2. Then

${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1-x}{1-xy}}(-\log(xy))^{s}\,dx\,dy=\Gamma (s+2)\left(\zeta (s+2)-{\frac {1}{s+1}}\right).}$

Here Γ is the Gamma function and ζ is the Riemann zeta function.

## Background

The first instance of the formula was proved and used by Frits Beukers in his 1978 paper giving an alternative proof of Apéry's theorem.[1] He proved the formula when s = 0, and proved an equivalent formulation for the case s = 1. This led Petros Hadjicostas to conjecture the above formula in 2004,[2] and within a week it had been proven by Robin Chapman.[3] He proved the formula holds when Re(s) > −1, and then extended the result by analytic continuation to get the full result.

## Special cases

As well as the two cases used by Beukers to get alternate expressions for ζ(2) and ζ(3), the formula can be used to express the Euler-Mascheroni constant as a double integral by letting s tend to −1:

${\displaystyle \gamma =\int _{0}^{1}\int _{0}^{1}{\frac {1-x}{(1-xy)(-\log(xy))}}\,dx\,dy.}$

The latter formula was first discovered by Jonathan Sondow[4] and is the one referred to in the title of Hadjicostas's paper.

## Notes

1. ^ Beukers, F. (1979). "A note on the irrationality of ζ(2) and ζ(3)". Bull. London Math. Soc. 11 (3): 268–272. doi:10.1112/blms/11.3.268.
2. ^ Hadjicostas, P. (2004). "A conjecture-generalization of Sondow's formula". arXiv:math.NT/0405423.
3. ^ Chapman, R. (2004). "A proof of Hadjicostas's conjecture". arXiv:math/0405478.
4. ^ Sondow, J. (2003). "Criteria for irrationality of Euler's constant". Proc. Amer. Math. Soc. 131: 3335–3344. doi:10.1090/S0002-9939-03-07081-3.