title: Hadjicostas's formula
text: # 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.
  
  