Integral Representation for Riemann-Siegel Z (t) function

We apply Poisson formula for a strip to give a representation of Z (t) by means of an integral. (t) =-^ h (x) (4+ix) 7-t{7}\, dx, Z (t) = F (t) (14+t²) ^{12 (254+t²) ^12}. \ After that we get the estimate (t) = (t2) ^74\e^{i (t) H (t) \}+O (t^-3/4), \ with (t) =-^ (t2) ^ix/2 (4+it+ix) 7 (x/7) \, dx= (t2) ^-74₍=₁^ 1n^{12+it}21+ (t{2 n²) ^-7/2}. \ We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.