# A conjecture about Xi function

1. Dec 11, 2009

### zetafunction

i've got the following conjecture about XI function, the following determinant

$$p_n(x) = \det\left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots & & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ 1 & x & x^2 & \cdots & x^n \end{matrix} \right]$$

with $$\mu _{2k}= \frac{a_{2k}}{a_{0}}(2k)!$$ for k even , if k=0 then is equal to 1

$$\mu _{2k+1}=0$$ for k odd

and $$\xi (1/2+iz)= a_{0} + \sum_{n=1}^{\infty}a_{2n}(-1)^{n}z^{2n}$$

tends to the xi function as $$2n \rightarrow \infty$$ (for n big and even integer)

the roots of the determinant are REAL and simple and are the roots for the xi function or at least asymptotically both set of roots

$$\frac{x_{2n}}{y_{2n}} =1$$ for big 'n' [/tex]

the idea behind this is that the xi function is somehow an 'orthogonal polynomial' of big degree 2n

Last edited: Dec 11, 2009