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In [[mathematics]], '''Hadjicostas's formula''' is a formula relating a certain [[Multiple integral|double integral]] to values of the [[Gamma function]] and the [[Riemann zeta function]].
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==Statement==
 
Let ''s'' be a [[complex number]] with Re(''s'')&nbsp;&gt;&nbsp;&minus;2. Then
:<math>\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).</math>
 
Here &Gamma; is the [[Gamma function]] and &zeta; 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]].<ref>{{cite journal|first1=F.|last1=Beukers|title=A note on the irrationality of ζ(2) and ζ(3)|journal=Bull. London Math. Soc.|volume=11|issue=3|year=1979 |pages=268&ndash;272|doi=10.1112/blms/11.3.268}}</ref>  He proved the formula when ''s''&nbsp;=&nbsp;0, and proved an equivalent formulation for the case ''s''&nbsp;=&nbsp;1. This led Petros Hadjicostas to conjecture the above formula in 2004,<ref>{{cite arxiv|first1=P.|last1=Hadjicostas|eprint=math.NT/0405423|title=A conjecture-generalization of Sondow’s formula|year=2004}}</ref> and within a week it had been proven by Robin Chapman.<ref>{{cite arxiv|first1=R.|last1=Chapman|eprint=math/0405478|title=A proof of Hadjicostas’s conjecture|year=2004}}</ref>  He proved the formula holds when Re(''s'')&nbsp;&gt;&nbsp;&minus;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 &minus;1:
 
:<math>\gamma=\int_0^1\int_0^1\frac{1-x}{(1-xy)(-\log(xy))}\,dx\,dy.</math>
 
The latter formula was first discovered by Jonathan Sondow<ref>{{cite journal|first1=J.|last1=Sondow|title=Criteria for irrationality of Euler's constant|journal=Proc. Amer. Math. Soc.|volume=131|year=2003|pages= 3335&ndash;3344|doi=10.1090/S0002-9939-03-07081-3 }}</ref> and is the one referred to in the title of Hadjicostas's paper.
 
==Notes==
{{reflist}}
 
==See also==
*{{cite arxiv
|first1=Kh.
|last1=Hessami Pilehrood
|first2=T.
|last2=Hessami Pilehrood
|title=Vacca-type series for values of the generalized-Euler-constant  function and its derivative
|eprint=0808.0410
|year=2008
}}
 
*Sondow, J. (2005). "[http://arXiv.org/abs/math.CA/0211148 Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula,]" ''[[American Mathematical Monthly]]'' '''112''': 61-65.
 
{{DEFAULTSORT:Hadjicostas's Formula}}
[[Category:Zeta and L-functions]]

Latest revision as of 10:37, 13 January 2015

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