# Integral equation for Xi-function

1. Nov 17, 2009

### zetafunction

think i have discovered an integral equation for the Xi-function

$$\Xi (z)= A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x+z) \frac{dx}{x}$$

with

$$\Phi(u) = \sum_{n=1}^{\infty}(2\pi ^{2} n^{4}e^{9u}-3\pi n^{2}e^{5u} )exp(-\pi n^{2}e^{4u})$$

and 'A' is a Real constant.

2. Nov 17, 2009

### zetafunction

of course i am referring to $$\Xi(z)= \xi (1/2+iz)$$ and due to the functional equation this Xi is even so we can formulate the integrale equation as

$$\Xi (z)= A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x+z) \frac{dx}{x}=A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x-z) \frac{dx}{x}$$
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