Talk:Riemann Xi function

Planet Math errors
The planet-math article implies that


 * The upper-case $$\Xi$$ function is defined as



\Xi(s) = \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s) $$


 * although Riemann himself used the notation of a lower case xi ($$\xi$$) for this expression. The famous Riemann Hypothesis is equivalent to the assertion that all the zeros of $$\Xi$$ are real, in fact Riemann himself presented his original hypothesis in terms of that function.

However, this is in direct conflict with mathworld, amnd mathworld appears to be more correct, in saying $$Xi(t)=\xi(1/2+it)$$ or something like that. This text should be fixed at mathworld, and fixe here. linas 20:56, 2 June 2006 (UTC)


 * This article is concerned with the lowercase xi function, but it is named using the upper case Xi. Either this article should say anything about Xi, or it should be moved to the name with lowercase xi. Oleg Alexandrov (talk) 23:42, 3 June 2006 (UTC)
 * The article now says something about Xi, nevertheless I think that it still should be moved to Riemann xi function, as the lowercase xi function is more important and better known. — Emil J. 13:23, 16 October 2009 (UTC)

Connes work?
I removed the following material from the article. In the current form, it is unreferenced and unintelligible.


 * As pointed by several works by Alain Connes and others, the Riemann hypothesis is equivalent to the assertion that the Riemann xi function is the functional determinant of the operator
 * $$ -D^{2}+f(x) \,$$


 * with
 * $$ f^{-1}(x) = \sqrt {4\pi} \frac{d^{\frac{1}{2}}N(x)}{dx^{\frac{1}{2}}}, \text{ so } \frac{\xi\left(\frac{1}{2}+iz\right)}{\xi\left(\frac{1}{2}\right)} = \frac{\det(H-z^{2})}{\det(H)} $$


 * this conjecture is supported by several numerical evaluations

AxelBoldt (talk) 02:10, 17 April 2012 (UTC)

Factor $$\frac12 s(s-1)$$
This article defines:


 * $$\xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s)$$

However, you can also see another definition:


 * $$\xi(s) = \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s)$$

The formula given for the value at the positive even integers:


 * $$\xi(2n) = (-1)^{n+1}\frac{n!}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n-1)$$

appears to confuse these two definitions for $$n>1$$. For example, is $$\xi(4)=\frac{\pi^2}{15}$$ or $$\xi(4)=\frac{\pi^2}{90}$$?

/80.71.142.78 (talk) 12:11, 19 May 2018 (UTC)