# Integral equation for Riesz function

1. Feb 24, 2010

### zetafunction

i was thinking about an integral equation for Riesz function

$$e^{-x}-1= \int_{0}^{\infty}\frac{dt}{t}frac(\sqrt (xt)Riesz(1/t)$$

apparently if should work , in fact if i take Mellin transform to both sides it gives me

$$\Gamma (s)= \frac{\Gamma(s+1)}{\zeta (-2s)} \frac{\zeta(-2s)}{s}$$

if i make the essay (by professor wolf) for the Riesz function $$Riesz(x)= x^{1/4}sin(A- \gamma /2 log(x))$$ , with 'gamma' being the imaginary part of the First non-trivial zero for the Riemman zeta function i get the result

0=1 ¡¡ , and i do not know how to follow.

Euler's formula $$2i sin(log(x))=x^{i}+x^{-i}$$

and the fractional part representation (valid for 1>s>0 )

$$-\zeta(s)=s\int_{0}^{\infty}dt frac(1/t)t^{s-1}$$