- #1

zetafunction

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

[tex] 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] [/tex]with [tex] \mu _{2k}= \frac{a_{2k}}{a_{0}}(2k)!

[/tex] for k even , if k=0 then is equal to 1

[tex] \mu _{2k+1}=0 [/tex] for k odd

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

tends to the xi function as [tex] 2n \rightarrow \infty [/tex] (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

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

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

[tex] 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] [/tex]with [tex] \mu _{2k}= \frac{a_{2k}}{a_{0}}(2k)!

[/tex] for k even , if k=0 then is equal to 1

[tex] \mu _{2k+1}=0 [/tex] for k odd

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

tends to the xi function as [tex] 2n \rightarrow \infty [/tex] (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

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

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

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