Hadjicostas's formula
# Hadjicostas's formula

In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the gamma function and the Riemann zeta function. It is named after Petros Hadjicostas.

## Statement

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

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

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. 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, and within a week it had been proven by Robin Chapman. 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:

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

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

