Riemann Xi function
In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Definition Riemann's original lower-case "xi"-function, was renamed with an upper-case (Greek letter "Xi") by Edmund Landau. Landau's lower-case ("xi") is defined as for . Here denotes the Riemann zeta function and is the Gamma function. The functional equation (or reflection formula) for Landau's is Riemann's original function, rebaptised upper-case by Landau, satisfies , and obeys the functional equation Both functions are entire and purely real for real arguments. Values The general form for positive even integers is where Bn denotes the n-th Bernoulli number. For example: Series representations The function has the series expansion where where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of . This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λn > 0 for all positive n. Hadamard product A simple infinite product expansion is where ρ ranges over the roots of ξ. To ensure convergence in the expansion, the product should be